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HARMONIC  STRUCTURE 


AND 


ELEMENTARY  COMPOSITION 


SEEGER  AND  STRICKLEN 


HARMONIC  STRUCTURE 


AND 


ELEMENTARY  COMPOSITION 


AN    OUTLINE    OF    A    COURSE    IN 
PRACTICAL   MUSICAL   INVENTION 


CHARLES  LOUIS  SEEGER,  Jr. 

(Professor  o(  Music  in  the  University  of  California) 
AND 

EDWARD  GRIFFITH  STRICKLEN 

(Instructor  in  Music  in  the  University  of  California) 


Revised  by  Edward  Griffith  Stricklen  from  "An  Outline 

of  a  Course  in  Harmonic  Structure  and 

Simple  Musical  Invention" 


4  r  J  5 1 

BERKELEY 
1916 


COPYRIGHT.  1915 

By  Charles  Louis  Seeger,  Jv. 

and 

Edward  Griffith  Stricklen 


MUSIC 
LIBRARV 


INTRODUCTION 


It  is  becoming  more  and  more  of  an  accepted  fact  that  far  from  being  opposed  and  mutually  exclu- 
sive, as  some  still  hold,  the  composition,  i)erformance  and  appreciation  of  music  are,  in  reality,  inter- 
dependent and  correlative  activities  constituting  the  three  most  important  functions  of  a  larger  unit — 
namely,  the  living  art  of  music.  An  isolation  similar  to  that  experienced  by  the  art  as  a  whole,  from 
any  obvious  connection  with  the  ebb  and  flow  of  social  and  intellectual  life  of  recent  years,  may  be 
noted  in  the  relations  of  the  three  activities  mentioned  not  only  to  one  another,  but  to  the  whole  art ; 
and  it  is  the  belief  of  the  authors  of  the  present  volimie  that  much  may  be  gained  through  a  more  pre- 
cise understanding  of  any  one  of  them  in  terms  of  any  other. 

In  aim  and  scope  this  work  is  fitted  primarily  to  the  reconciliation  of  the  creative  and  appreciative 
faculties,  first,  by  means  of  explanation,  within  reasonable  limits,  of  the  logic  of  harmonic  structure — as 
well  to  the  benefit  of  the  philosophic  enquirer  as  to  the  alert  young  mind :  second,  through  an  under- 
standing, from  actual  experience  in  composition,  of  the  principles  underlying  the  European  art;  and 
third,  by  the  instruction  of  the  young  composer,  who  may  as  well  (as  any  one  will  grant)  be  aware 
of  the  modern  spirit  and  his  relationship  to  it,  whether  or  not  he  intends  trusting  his  muse  to  a 
guiding  intellect,  a  truant  fancy  or  to  somewhat  of  both.  For  whether  the  musician  of  the  older 
school,  who  relies  upon  "pure  instinct"  or  taste  for  his  authority,  likes  it  or  not,  one  can  no  longer 
Ignore  the  persistent  demand  from  a  younger  generation,  born  and  bred  in  an  age  of  increasing  scien- 
tific activity,  for  reasons  and  explanations  instead  of  the  dumb  rules  and  empirical  subterfuges  which 
ill  no  great  living  composer  are  regarded:  further,  one  cannot  rernain  blind  to  the  forces  which  set 
art  in  motion  or  to  the  forces  art  sets  in  motion,  because,  together  with  the  seemingly  last  gasps  of 
romantic  phantasy  has  appeared  a  tenuous,  at  least,  scientific  explanation  for  many  of  the  foibles 
hitherto  most  jealously  guarded  by  the  artistic  temperament  as  strictly  private  property. 

The  human  intelligence  has  suited  itself  with  unusual  consistency  to  the  ordering  of  the  main  ele- 
ments on  musical  com])osition — Rhythm,  Tone  and  l~orm — in  that  it  can  be  shown  with  convincing 
proof  that  the  underlying  laws  of  taste  and  nature  have  agreed,  as  to  their  beginnings  at  least,  upon 
one  fundamental  scheme  for  all  three. 

The  musical  means  for  the  Forming  of  Rhythm  and  Tone  into  Music  rests  upon  the  employment 
of  two  distinct  and  complementary  factors — Melody  and  Harmony.  It  has  been  a  widely  discussed 
question,  which  to  regard  as  the  more  fundamental  and  which  to  advocate  as  the  subject  better  suited 
for  presentation  to  the  student  entering  upon  a  musical  education.  In  older  days,  the  art  of  counter- 
])oint  (melofly  and  the  combination  of  melodies)  was  the  only  one  accepted  as  the  fundamental  train- 
ing, but  of  recent  years  (the  last  few  centuries)  the  instruction  has  been  first  in  "harmony"  (or,  more 
correctly  as  we  shall  see,  "applied  harmony").  It  has  become  apparent  to  the  authors  that  on  the  one 
hand,  the  art  of  counterpoint  has  been  .-uid  still  is  pincticable  only  upon  the  basis  of  a  definite  concep- 
tion of  harmonics,  while,  on  the  other,  harmonic  tendencies  :nirl  combinations  only  become  musical 
material  when  connected  according  to  the  accepted  tradition  of  melodious  part-writing. 

The  plan  has  developed,  therefore,  not  to  slight  one  factor  while  pretending  to  cultivate  style 
a|>art  from  it  by  means  of  the  other  alone  (a  thing  which  has  never  been  actually  carried  out")  but  to 
start  the  stu<lent  at  the  initial  steps  of  each  at  one  and  the  same  time,  thus  cutting  out  a  lot  of  dessi- 
cafing  routine  and  eventually — now-a-days — an  altogether  cataclysmic  reconciliation. 

Once  the  teacher  and  student  succeed  in  divesting  themselves  of  the  superstition  of  any  extraor- 
dinary difTiculty  in  music  reading,  writing,  grammar  and  rhetoric  and  of  the  conviction  of  its  esoteric 
intangilMlity  and  uselessness,  much  will  be  accomplished:  for  the  main  obstacles  encountered  in  musical 
studies  are  not  often  due  to  inherent  iinniusicalness.  but  rather  to  tardiness  in  undertaking  such  ele- 
mentary   work    and    to   preposterous    ideas    of    so-called  "llieor\-."  which  is  not  theory  at  all.  but  only 


stereotyped  practice.  Tliis  particular  misconception  lla^,  more  than  any  other,  served  to  cheapen  the 
study  of  the  art.  That  there  may  be  a  science  of  harmony  according  to  which  harmonic  laws  may  be 
deduced,  as  well,  also,  as  a  logic  to  the  artistic  method,  the  authors  believe  they  have  good  claim  to 
have  outlined ;  but  it  does  not  follow  that  practical  rules  may  be  given  for  the  operation  of  physical 
laws  or  that  theoretical  tenets  may  be  dragged  in  to  defend  contrapuntal  style.  As  to  this  department 
of  the  work,  the  authors  have  reason  to  believe  that  some  points,  such  as  the  logic  of  the  minor  mode, 
of  secondary  triads  and  seventh  chords,  etc.,  as  well  as  the  general  method  of  arranging  the  whole, 
are  original  with  them  ;  and  the  student  of  philosophic  or  scientific  bent  is  especially  invited  to  examine 
closely  to  the  strict  application  of  harmonic  theory  to  musical  details  in  the  beginning  of  the  course  and 
then  follow  its  slow  but  persistent  passage  to  more  general  aspects  according  as  the  fitting  to- 
gether of  the  reliques  of  ancient  tradition  and  modern  thought  becomes  more  and  more  difficult. 
That  a  certain  type  of  mind  craves  just  this  stimulus  and  that  other  types  stand  in  apparent  need  of  it, 
has  been  proven  again  and  again  within  the  authors'  experience ;  for  the  modern  spirit  is  becoming 
evident  in  musical  thought  in  an  unusually  concrete  fashion,  and  ideas  long  current  in  literary  and 
scientific  work  are  being  projected  into  the  chaotic  waters  of  modern  musical  life  which  may  lead 
upward  and  onward  in  true  Hegelian  fashion,  to  higher,  more  complete  and  more  comprehensive  forms 
— not  forsaking  the  old,  but  reaffirming  transcending  and  embellishing  it  as  each  new  fragment  of  dis- 
sonant chaos  is  conquered  and  found  beautiful  to  the  eyes  of  a  more  imiversal  consciousness. 

Berkeley,  California,  C.  L.  S.,  Jr. 

Tulv.  1016. 


PREFACE  TO  THE  SECOND  EDITION. 

The  work  represented  by  the  second  edition  is  divided  into  two  parts  of  fifteen  chapters  each,  and 
is  so  designed  that  a  semester  of  nineteen  weeks  or  thereabout  may  be  devoted  to  each  part,  including 
the  necessary  introduction,  review  and  examination. 

The  student's  preparation  should  include  such  training  as  will  enable  him  to  recognize  both  by 
notation  and  by  sound  the  signature  and  structure  of  all  scales,  intervals  and  metrical  types  and  a  few 
of  the  simpler  types  of  chords.  Familiarity  with  a  keyboard  instrument,  while  not  indispensable,  will 
be  nuich  to  his  advantage.  It  is  quite  essential  that  he  should  have  a  clear,  correct  and  neat  musical 
hand-writing  and  some  proficiency  in  melody-writing. 

For  the  cultivation  of  the  feeling  for  correct  and  melodious  voice  leading,  it  is  recommended  that 
studies  in  strict  counterpoint  be  carried  on  concurrently  with  the  work  in  Harmony.  The  authors' 
leaching  experience  has  proved  that  this  will  allo\v  ample  compensation  for  that  neglect  of  the  mel- 
odic element  of  music,  which  is  unavoidable  in  the  first  stages  of  the  study  of  harmony.  Of  the 
many  good  books  on  the  subject  C.  W.  Pierce,  "Modern  Academic  Counterpoint"  (G.  Schirmer,  New 
York)  has  been  found  most  useful. 

On  account  of  the  excellence  and  availability  of  Dr.  Pierce's  book,  no  exercises  in  Counterpoint  are 
included  in  the  present  volume.  It  seems  best  to  devote  the  first  term  to  the  study  of  Part  I  and  the 
hve  species  of  Counterpoint  in  two  parts  and  the  second  term  to  Part  II  and  Counterpoint  in  three 
I'arts,  although  the  complete  study  of  the  combination  of  species  in  three  parts  can  hardly  be  made 
until  later.  A  plan  for  further  studies  in  Counterpoint  will  be  indicated  in  the  preface  to  the  succeeding 
\olunie  on  "Chromatic   Harmony"    ( E.   G.    Strickleu). 

The  authors  believe  that  the  student  will  gain  the  most  solid  culture  in  music  by  the  steady 
increase  in  his  power  of  invention  through  sustained  effort  to  apply  to  his  own  work  each  detail  of 
knowledge  acquired.  For  this  reason,  melodies  to  be  harmonized  are  not  assigned.  The  teacher 
may  feel,  however,  that  the  needs  of  his  particular  problem  demand  the  supplying  of  such  melodies; 
hut  no  attempt  is  made  here  to  offer  a  general  solution  for  the  number  of  teaching  problems  of  which 
the  authors  are  well  aware. 

Xo  course  in  harmony  can  possess  educational  value  unless  its  relation  to  the  living  art  of  music 
is  always  apparent.  The  student  should  therefore,  be  as  closely  as  possible  in  touch  with  good  music 
throughout  his  course  in  composition  and  should  erideavor  to  observe  in  it  the  varied  use  of  the  mu- 
sical material  which  he  is  studying.  He  should  regard  the  steady  growth  of  his  own  musical  vocabu- 
lary both  in  the  light  of  increasing  opportunity  for  self-expression  and  of  increasing  power  to  appre- 
ciate the  beauty  of  the  best  examples  of  musical  composition.  E.  G.  S. 
Rerkelcv,  lOir,. 


PART  I. 


THE  DIATONIC  CONSONANCES. 
Chapter  One. 


PRIMARY    TRIADS. 


THE    MAJOR    SCALE. 


0.  A  science  of  harmony  may  be  based  upon  certain  physical  laws  deduced  from  phenomena  observed 
in  the  production  of  tone. 

The  result  of  setting  in  vibration  a  sounding  body  (for  example,  a  violin  or  pianoforte  string) 
seems  at  first  to  be  only  a  single  tone.  But  a  keener  observation  will  result  in  the  perception 
of  other  tones  which  are  present  during  the  sounding  of  the  string.  It  has  been  found  that 
certain  of  these  other  tones  (or  overtones)  arc  due  to  the  vibration  of  fractional  parts  of  the 
string,  wliich  take  place  at  the  same  time  as  the  vibration  of  the  string  as  a  unit.  Each  of  these 
fractional  parts  gives  a  different  tone  of  its  own  and  the  series  of  partial  tones  so  formed  extends 
indefinitely  upwards.  The  initial  degrees  of  a  normal  harmonic  series  are  shown  in  the  follow- 
ing illustration   (C  has  been  taken  as  a  convenient  starting  point)  : 


i? 

i T~ 

r1 
lr=i 

'k    'h   4- 

1   J      1^    ^              II 

The  numerals  printed  above  and  below  the  overtones  represent  the  fractional  parts  of  the 
sounding  body  (C)  which  produces  them.  The  lowest  C  is  called  the  first  partial  or  funda- 
mental; the  octave  (J'z),  which  vibrates  twice  as  fast,  the  second  partial;  the  twelfth  (^), 
which  vibrates  three  times  as  fast,  the  third  partial,  and.  so  on. 

b.  The  student  should  carefully  note  the  relations  of  the  constituents  of  this  harmonic  series  laoth  to  the 
fundamental  and  to  each  other,  for  they  are  typical  of  the  normal  harmonic  series  upon  any 
given  fundamental.  An  easy  inethod  of  memorizing  these  relations  is  to  note  that  between 
the  fundamental  and  the  first  overtone  is  a  perfect  octave,  between  the  first  and  second  over- 
tones is  a  perfect  fifth,  between  the  second  and  third  a  perfect  fourth,  and  so  on. 

(If  an  "open  string"  of  a  violin  or  a  violoncello  be  divided  in  one  or  the  other  of  the  frac- 
tional parts  noted  in  Figure  1,  it  will  be  found  that  the  tone  produced  by  this  fractional  part 
will  bear  the  same  relation  to  the  tone  produced  by  the  "open  string"  as  exists  between  the 
fundamental  and  the  similar  overtone  in  Fi^nre  1.) 

c  We  may  now  observe  that  Figure  1  is  composed  entirely  of  the  notes  C,  E,  and  T,,  which,  when 
more  simply  expressed,  as  in  I'igure  2  belov^',  coincides  with  one  of  the  simplest  and  most 
commonly  used  chords  and  undoubtedly  accounts  in  n  large  measure  for  our  choice  of  it  in 
amplifying  the  key  of  C  by  fuller  harmony. 

%.2. 


^^ 


Chapter  One  1 

il.  Our  studies  in  harmony  may  be  divided  under  the  two  following  heads:  (1)  the  origin  and  the 
derivation  of  chords,  and   (2)   their  relations  to  each  other. 

It  has  been  found  from  experience  that  the  relation  between  chords  depends  very  largely 
upon  the  relation  between  the  tones  which  generate  them.  In  this  respect,  we  may  call  the  tone 
C  the  generator  of  the  C  major  chord,  or  triad. 

By  examining  the  relation  between  the  fundamental  and  the  second  partial  in  Figure  1,  we 
may  conclude  that  the  simplest  relation  between  any  two  tones  is  that  of  a  perfect  octave,  for, 
when  the  fundamental,  taken  as  a  unity,  is  compared  to  the  overtones  (which  may  be  ex- 
pressed as  fractions)  our  musical  sense  tells  us  that  the  second  and  fourth  partials  are  to  a 
large  degree  only  reflections  or  reduplications  in  miniature  of  the  original  fundamental,  where- 
as already  in  the  third  partial  a  decided  feeling  of  difference  is  immediately  established.  The 
relation  between  the  fundamental  and  the  next  partial  above  the  fourth  will  be  seen  to  be  more 
complex  than  that  between  the  fundamental  and  the  third  partial.  We  say,  therefore,  that  the 
relation  1 :3  is  the  simplest  relation  possible  between  two  different  tones,  and,  applying  the 
above  remark  regarding  octaves,  treat,  for  the  present,  the  perfect  twelfth  as  a  perfect  fifth 
by  transposition. 

The  following  figure  gives  the  note  C  and  the  notes  F  and  G  which  will  be  seen  to  be  in 
fifth   relation  to  it. 


ffUfc  ^  \ 


7i^ 


m 


zt: 


e.  Since  every  tone  produces  a  major  triad  from  its  overtones,  we  may  proceed  to  erect  one  on  each 
of  the  notes  of  Figure  3,  as  illustrated  below. 


If  the  notes  in  the  above  figures  are  arranged  in  an  ascending  line  from  the  note  C  we  will 
obtain  the  scale  of  C  major  as  in  the  following  illustration. 


^^ 


In  the  above  illustration,  the  chords  of  Figure  4  are  placed  above  their  generators  as  found  in 
the  scale. 
/.  Names  of  the  degrees  of  the  scale:     1,  Tonic;  2.  Supertonic:  3,  Mediant;  4,  Subdominant;  5,  Domi- 
nant ;  6,  Submcdiant ;  7,  Leading  tone. 

As  a  chord  is  always  named  after  the  tone  from  which  it  is  built.  Figure  5  presents  the 
triads  of  the  Tonic,  Subdomiiinnt.  and  Dominant. 


g  Chapter  Tu-o 

FXERCISE. 

Write  and  play  major  scales  beginning  on  the  following  different  tones,  above  middle  C, 
erecting  first  a  harmonic  series  upon  each;  G,  D,  A,  E,  B,  F  sharp. 

Below  middle  C :  F,  B  flat,  E  flat,  A  flat,  D  flat,  G  flat.  Erect  the  Tonic,  Dominant,  and  Sub- 
dominant  triads  in  each  key — thirteen  in  all. 

E.\R    TR.MNING. 

Learn    to    distinguish    the    primary    triads  when  heard. 


Chapter  Two. 

CHORD   CONNECTION. 

((.  From  Figure  4  of  the  preceding  chapter  we  perceive  the  Tonic  triad  to  be  graphed  as  the  center  of 
a  key  system.  The  musical  faculty  leads  us  to  accept  this  chord,  as  a  point  of  repose,  as  the 
normal  harmony  for  beginning  or  ending  an  ordinary  composition.  The  student  will  find  him- 
self already  familiar  with  the  three  triads  of  Figure  4  as  being  the  chords  most  generally  used 
in  establishing  the  key  of  C  major. 

b.  As  the  Tonic  triad  is  the  point  of  repose  of  a  key,  its  tones  will  in  melodic  treatment  tend  to  pass  to 
other  tones  of  the  scale  and  these  other  tones  to  pass  back  to  the  tones  of  the  Tonic  triad.  This 
movement  is  unrestricted  when  the  progression  is  made  away  from  the  Tonic  triad,  but,  when 
made  back  to  it,  the  most  obvious  tendency  of  any  tone  not  a  member  of  the  Tonic  triad  is  to 
proceed  to  the  nearest  tone  of  this  chord  as  a  point  of  repose. 

On  account  of  these  tendencies,  the  second  fourth,  si.xth.  and  seventh  scale-degrees  are  called 
"ACTIX^E  TONES."  Below,  in  Figure  6,  the  members  of  the  Tonic  triad  (sometimes  called 
"INACTIVE  TONES")  are  given  in  "white"  notes,  and  the  Active  Tones  in  "black"  notes,  the 
curved  arrows  indicating  the  "tendencies"  of  the  active  tones. 


g 


=^5^ 


^^?^ 


A  consideration  of  the  ideas  brought  forward  in  the  preceding  section  gives  us  the  first  rules  for 
the  practice  of  chord  connection. 

The  simplest  way  of  applying  them  will  be  to  remember  that,  for  the  present,  we  should  (1) 
begin  and  end  each  e.xercise  on  the  Tonic  chord.  (2)  we  should  lead  the  active  tones  according 
to  their  recognized  tendencies.  At  present,  the  Tonic  triads  may  contain  any  one  of  their 
three  notes  in  the  lowest  part ;  the  form  of  the  other  chords  will  depend  upon  their  context. 

It  will  produce  a  better  musical  "texture"  if  the  inner  part  is  kept  within  the  distance  of 
an  octave  from  the  two  outer  ones,  and  if  common  tones  are  tied  over  in  the  same  part.  Fig- 
ure 7  below  is  an  exerci.se  constructed  in  accordance  with  these  principles. 


f     I'    '  f     f 


Chapter  TJiree  9 

Students  should  carefullj-  analyze  this  exercise,  giving  particular  attention  to  the  spacing  of 
each  chord,  the  tying  of  each  common  tone,  and  the  conduct  of  each  active  tone. 

For  the  present,  the  best  method  for  working  out  exercises  may  be  said  to  be  the  following: 

(1)  Select  the  key  and  write  its  Tonic  triad  in  one  of  the  several  arrangements  that  will 
occur  to  the  mind ; 

(2)  Proceed    to    either    the    Dominant    or  Subdominant  triad,  first  tying  the  common  tone 
and  then  leading  the  other  parts  as  smoothly  as  possible ; 

(3)  Return  to  the  Tonic  triad,  first  tying  the  common  tone  and  then  following  the  tendency 
of  the  active  tones. 

d.  It  is  generally  found  convenient  to  name  chords  by  the  number  of  the  scale-degree  on  which  they 

are  built.     Referring  to  Chapter  One,  Figure  5,  we  see  that  we  may  allude  to  the  Tonic  chord 
as  the  I,  to  the  Subdominant  chord  as  the  lY,  and  so  on. 

e.  The  student  may  invent  his  own  exercises  in  chord  connection  from  the  knowledge  already  gained. 

It  will  be  better  for  him  to  avoid,  at  present,  the  progressions  IV-V  and  V-IV. 

INVENTION. 

After  the  above  directions,  invent  exercises  in  chord  connection,  of  various  lengths,  in  vari- 
ous major  keys,  using  different  varieties  of  simple  measure. 
Figure  7  may  be  taken  as  a  model. 
/.  The  rules  for  the  treatment  of  active  tones  may  be  stated  as  follows : 

(1)  As  the  second  degree  lies  between  the  first  and  third,  it  may  be  led  to  either,  but  at 
present  had  better  be  led  to  the  third  only. 

(2)  Lead  the  fourth  degree  to  the  third. 

(3)  Lead  the  sixth  degree  to  the  fifth. 

(4)  Lead  the  seventh  degree  to  the  eighth. 


Chapter  Three. 


FOUR   P.\RT  H.VRMOXV. 


a.  When  a  triad  occurs  in  the  form  given  in  Figures  4  or  5  of  Chapter  One,  it  is  said  to  be  in 
"ROOT  POSITION."  The  lowest  member  is  called  the  Root,  the  middle  tone  is  called  the 
Third,  the  highest  is  called  the  Fifth.  These  names  are  retained,  no  matter  in  what  location  the 
tone  so  named  may  occur. 

/).  From  consideration  of  l-'igure  1,  Chapter  One,  it  will  be  seen  that  the  most  important  member  of  a 
major  triad  is  its  root.  Consequently,  if  we  desire  to  write  a  major  triad  in  four  parts,  par- 
ticularly in  root  position,  the  root  will  always  be  the  best  member  to  double.  Figure  8  below 
gives  various  examjiles  of  the  C  major  triad  in  root  position  "doubled"  after  this  principle. 
Observe  that   the   duiilication  can  take  place  in  any  one  of  the  upper  parts. 

%.5 


s 


s 


10 


Chapter  Four 


c.  In  four  part  harmony,  tlie  lowest  part  is  called  the  Bass,  the  part  above  it  the  Tenor,  the  next 

higher  part  the  Alto,  and  the  highest  part  the  Soprano. 

d.  The  spacing  of  the  three  upper  parts  should  be  the  same  as  that  employed  for  the  three-part  work  in 

the  preceding  chapter.    A  wider  spacing  between  Tenor  and  Bass  may  be  employed  when  it 
seems  necessary. 

EXERCISE. 

Build  up.  in  four  parts,  five  different  arrangements  of  each  of  the  three  known  chords,  in 
six  different  keys,  using  root  position  only. 

As  this  is  not  an  exercise  in  chord  connection,  each  chord  should  be  written  as  a  detached 
example  of  correct  doubling,  as  in  Figure  8. 

c.  On  account  of  their  great  value  in  determining  a  key,  these  three  chords  are  known  as  the  "PRIN- 
CIPAL or  PRIMARY  TRIADS."  The  relation  that  they  bear  to  the  key-note,  shown  in 
Figure  4,  Chapter  One,  further  explains  this  name. 

We  may  still  call  the  chords  of  Figure  8  "triads,"  because  the  fourth  part  is  merely  a  duplica- 
tion on  another  level  of  a  note  already  present. 

NOTE — Avoid  confusion  between  the  third  degree  of  a  scale  and  the  third  of  a  chord.  Re- 
flection will  show  that  these  notes  are  the  same  in  one  case  and  different  in  others. 


Chapter  Four. 


CHORD  CONNECTION    IX   FOUR   PART.S. 


a.  The  invention  given  in  Chapter  Two  may  be  now  worked  out  in  four  parts  in  the  following  manner : 

(1)  Place  the  root  of  each  triad  in  the  Bass  before  writing  the  other  members  of  the  chord. 

(2)  Follow  the  given  directions  for  the  treatment  of  active  tones  and  spacing. 

(3)  Avoid,  at  present,  the  tying  of  a  common  tone  in  the  Bass. 

(4)  Write  the   Alto   and   Soprano   on   the  treble  staff,  and  the  Tenor  and  Bass  on  the  bass 
staff. 


Figure  9  below  gives  an  exanii)le  of  such  work. 


^ 


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s 


r 


32= 


-0^ 


Chapter  Four 


11 


b.  We  may  now  include  in  our  work  the  four  part  progresion  IV-V.  There  is  no  common  tone 
present,  but  a  satisfactory  treatment  of  the  active  tones  will  give  a  good  result.  This  may 
be  illustrated  in  the  following  figure. 


^xa.lO 


(T):      0 


Note  that  the  root  of  the  Subdominant  triad,  in  the  Bass,  is  not  treated  as  an  active  tone. 
This  is  due  to  its  greater  importance  as  the  root  of  the  Subdominant  chord.  Note  that  the 
doubled  root  of  the  Subdominant  in  the  Soprano  receives  only  approximately  the  usual  active 
tone  treatment,  as  the  tone  to  which  it  generally  moves  is  not  present  in  the  following  chord. 
The  rules  for  this  chord  progression  may  be  summarized  as  follows: 

Lead  the  root  of  the  Subdominant  to  the  root  of  the  Dominant ;  lead  the  doubled  root  of  the 
Subdominant  to  the  fifth  of  the  Dominant ;  lead  the  third  of  the  Subdominant  to  the  doubled  root 
of  the  Dominant ;  lead  the  fifth  of  the  Subdominant  to  the  third  of  the  Dominant. 

c.  Avoid  for  the  present  the  progression  V-IV.     It  has  generally  an  unpleasant  effect,  in  which  the 
student  of  Counterpoint  will  recognize  the  forbidden  "Tritone." 

EXERCISE. 

Write  each  of  the  following  chord  sequences  in  several  different  major  keys  according  to  the 
given  directions.     \^arious  forms  of  measure  should  be  used. 

(1)  I-IV-V-I. 

(2)  I-V-MV-I-IV-V-I. 
(3)I-I-IV-IV-V-V-I. 

The  third  exercise  here  follows  in  the  key  of  C  major.  Note  that  when  a  chord  is  repeated 
its  tones  may  be  freely  rearranged,  whether  they  are  active  tones  or  not. 


^^•"       I       I       IT      IF     ^     -^ 


P 


i^S 


t=f 


^g 


4r 


i 


^^ 


3i^ 


4fc 


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— o: 


INVENTION. 

Invent  six  chord  sequences,  similar  to  those  in  (2)  and  (3)  above,  in  various  major  keys  and 
meters  as  before.  Always  begin  and  end  with  the  Tonic  triad,  and  have  each  cliord  in  root 
position. 


Chapter  Five 


Chapter  Five. 

THE    MINOR    SCALE    AND    PRIMARY    TRIADS    IN    THE    MINOR    MODE. 

(The  student  should  compare  each  section  of  the  present  chapter  with  the  similarly  lettered 
sections  of  Chapter  One.) 


^^.  IX/    -^ 


5  p      TLt>     ,{.¥■ 


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i^6 


a.  If  the  first  note  G  of  I'^igure  12  be  taken  as  a  new  point  of  departure,  and   the  length  of  a   string 

capable  of  producing  it  is  doubled,  the  note  resulting  from  this  doubling  will  be  a  [)crfect 
octave  below  it,  as  shown  in  the  figure.  Similarly,  multiplying  the  original  string-length  by 
three,  by  four,  by  five,  and  by  six,  tones  will  be  obtained  of  the  same  pitch  as  those  shown  in 
the  figure.  On  account  of  their  location  in  pitch,  the  tones  thus  formed  below  the  original  note 
G  are  called  Undertones. 

b.  By  a  comparison  of  Figure  12  with  Figure  1.  the  student  will  observe  that  the  series  of  undertones 

presents  the  same  series  of  interval-relations  to  their  generator  and  to  each  other  as  e.xists  be- 
tween the  fundamental  and  the  overtones  of  Figure  1,  the  only  difference  being  that  while 
overtones  are  projected  upward,  undertones  are  projected  downward. 

c.  We  may  now  observe  that  Figure  12  is  composed  entirely  of  the  notes  G,  E  flat,  and  C     This  fact 

may  account  for  the  existence  of  the  minor  triad,  which  may  be  more  simply  exprei.sed  as  in 
Figure  13  below. 


^. 


^3 


d.  By  substituting  the  word  "undertone"  for  "overtone"  in  section  d  of  Chapter  One,  we  will  find  that 
the  assertion  concerning  the  relationship  of  the   perfect  fifth  holds  true  here  also. 

Applying  this  fact,  we  obtain  the  following  figure  which  consists  of  the  note  G  and  the  notes 
C  and  D  which  are  in  fifth  relation  to  it  below  and  above,  respectively. 


m 


c.  Since  from  any  tone  a  triad  composed  of  undertones  may  be  projected,  we  may  now  build  such  triads 
on  each  of  the  notes  of  Figure  14  with  the  following  result: 


Chapter  Five 


13 


/.  Observe  thai,  in  Chapter  One,  the  overtones,  triads,  and  the  resulting  scale  are  all  produced  upwards. 
Now,  in  the  illustrations  of  the  present  chapter  we  may  note  that  the  undertones  and  the  chords 
may  be  built  downwards.  Following  out  this  thought,  we  may  take  the  chords  of  Figure  15, 
and,  beginning  with  the  note  G,  arrange  their  members  in  a  descending  series  with  the  following 
result : 


^•■'^ii      °    ^"   b§     LH     ^0    Lo 


This  is  the  true  or  theoretical,  minor  scale  and  we  call  the  chord  G  E  flat  and  C  the  Tonic 
triad  of  the  minor  mode. 
(/.  But  our  modern  practice  conceives  of  chords  as  ])rojected  upward  from  the  bass.    In  this  illustration, 
C,  both  by  analog^'  to  its  position  in  the  majoi-  triad  and  as  a  result  of  contrapuntal  practice,  is 
taken  as  the  root  of  the  chord,  while  the  scale  is  thence  described  upwards  from  it.* 

The  notes  and  chords  of  Figure  16  rearrange  themselves  therefore  in  the  following  order: 


% 


■  n 


L„    ^8     ^t    ^0 


ia- 


Observe  that  the  chords  on  the  fourth  and  fifth  degrees  of  this  scale  occupy  the  reverse 
position  with  respect  to  the  original  Tonic  chord  than  that  occupied  in  Figure  16.  This  fact 
is  of  the  utmost  importance  in  later  studies  of  the  harmonies  of  the  minor  key. 
On  account  of  the  frequent  necessity  for  the  intensifying  effect  of  the  Leading  Tone  as  found 
in  the  major  mode,  the  seventh  degree  of  the  scale  of  Figure  17  is  frequently  raised  as  in  the 
following  figure : 


^■-iX  .  \i 


m 


ifcM 


1 


This  results  in  presenting  two  available  forms  for  the  triad  on  liie  fiftii  degree  of  Figure  17, 
i.  c,  as  minor  or  as  major.     Roth  of  these  forms  will  be  foimd  in  use. 
I.  The  scale  given  in  I'igure  18  with  the  raised  Leading  Tone  is  the  familiar  "Harmonic  Minor  Scale." 

Its  degrees  and  its  chords  are  named  as  in  Chapter  One,  section  f.    When  the  raised  Leading 
Tone  is  not  used,  it  is  called  the  "natural  seventh." 

l■:XIiRCISI^. 

.\l)p!y  the  material  of  the  minor  key  to    the  e.xercises  of  Chapter  One.  including  both  forms 
of  the   Dominant  chord. 

I    \l<    TR.MNINC. 

Learn  to  recognize  the  difference  between  the  major  and  minor  scales  and  chords. 

•European  musical  tradition  has,  since  mediaeval  times,  consistently,  both  in  theory  and  in  practice,  measured 
intervals  from  the  bass  upwards ;  this,  together  with  the  fact  that  harmonic  series  in  superior  resonance  seem  to  be 
commoner  phenomena  in  our  experience  than  harmonic  series  in  inferior  resonance,  has  displaced  the  logical  center  of 
the  mode  for  so  lonp.  that  the  accepted  styles  of  the  present  day  regard  the  mode's  dominant  (C)  as  the  tonic;  and. 
under  influence  of  the  m.ijor  mode  (whicli  recommends  itself  to  us  as  the  more  normal  idiom),  many  of  its  character- 
istics have  remained  unrecognized  wliile  others  have  been  distorted,  the  resulting  anomaly,  like  the  other  modes  of 
the  medijpval  church,  seem  only  tolerably  fitted  for  harmonic  treatment  in  view  of  the  strong  feeling  for  KEY  devel- 
oped in  recent  years. 


14  Chapter  Six 

Chapter  SUv. 

CHORn    CONNECTION    AND    FOUR    PART    HARMONY    IN    THE    MINOR    MODE. 

a.  For  similar  reasons,  the  rules  for  active  tones  and  chord  connection,  given  in  Chapter  Two,  apply 

to  the  connection  of  the  chords  of  the  minor  mode. 

b.  We  apply  the  names  of  Root,  Third  and  Fifth  to  the  minor  chords  as  we  did  in  the  major  triad. 

c.  For  these  reasons,  we  double  the  Root  of  a  minor  triad  when  writing  it  in  four  parts  in   Root 

position,  treating  it  in  this  case  exactly  as  we  would  in  the  major. 

Sections  c  and  d  of  Chapter  Three  apply  to  the  minor  mode  as  well,  as  does  also  section  e 
of  the  same  chapter,  for  the  same  reason  there  given. 

KXERCISE. 

Work  out  the  exercise  of  Chapter   Three  in   six  different  keys,  in  the  minor  mode. 

(/.  The  entire  material  of  Chapter  Four  may  now  be  applied  to  the  chords  of  the  minor  mode,  in 
various  keys. 

To  this  we  may  add  that  the  progression  V-IV  in  the  minor  mode  may  be  freely  allowed 
when  V  is  taken  as  a  minor  triad,  there  being  as  a  rule  no  bad  Tritone  effect.  \\'hen  \'  is  a 
major  triad  the  progression  is  as  bad  as  the  similar  triad  in  major,  and  for  the  same  rea.sons. 

e.  The  Dominant  as  a  major  triad  should  not  be  neglected,  however,  and  if  the  exercise  closes  with 
the  progression  V-I,  \'  should  be  a  major  triad  if  the  effect  of  the  raised  Leading  Tone  sounds 
better. 

EXERCISE. 

Work  out  the  exercise  of  Chapter  Four  in  various  keys,  using  the  minor  mode.  The 
treatment  is  the  same. 

INVENTION. 

Work  out  the  invention  of  Chapter  i'our  in  various  keys,  using  the  minor  mode  and  follow- 
ing the  given  directions. 


Chapter  Seven. 

THE    FIRST    PRlNriPLES    or     MELODIC    INVENTION. 

(This  chapter  should  be  omitted  by  students  who  have  not  been  studying  Strict  Counter- 
point concurrently  with  this  course  in  Harmony.) 

a.  Melodies  cannot  be  made  by  rule,  but  all  melodies  follow,  in  a  general  sense,  .some  rational  course 

of  construction. 

b.  With   practice,   a  constantly  increasing  series  of  melodic  opportunities  will   be  discovered  by  the 

student.  He  will  find  his  best  guides  to  these  opportunities  to  be  (1)  a  careful  observation 
of  the  tendencies  of  all  active  tones,  (2)  a  consideration  of  the  harmonic  implication  of  each 
note  that  he  writes,  and  (3)  a  careful  choice  of  the  duration  of  the  tones  he  employs  with 
respect   to  each  other.     The  following  comments  may  be  added  on  these  general  principles: 

( 1 )  -An  active  tone  should  always  be  led  according  to  its  tendency,  except  where  a  skip 
from  one  tone  to  another  seems  to  imply  that  both  tones  belong  to  the  same  chord.  A  freer 
treatment  of  this  principle  will   be  discussed  in  the  following  chapter. 


Chapter  Eiglit  15 

(2)  A  melody  should  be  so  conducted  that,  if  harmonized,  no  forbidden  harmonic  pro- 
gression may  occur.  All  melodies  should  begin  and  end  with  one  of  the  members  of  the  Tonic 
triad. 

(3)  For  the  present,  the  melodies  should  be  written  in  half-notes  and  quarter-notes.  A 
whole-note  may  be  used  as  the  last  note  of  a  melody,  when  the  exercise  is  written  in  4/4 
meter.  In  3/2  or  6/4  meter  whole-notes  may  be  more  freely  used.  Dotted  half-notes  or  even 
dotted  whole-notes  may  be  used  when  possible,  to  secure  rhythmic  variety.  For  the  present, 
it  will  be  better  to  introduce  the  longer  notes  of  a  melody  on  the  more  strongly  accented  beats 
of  the  measure. 

c.  Skips  should  be  used  as  is  Counterpoint  and  should   follow  the  same  rules. 

</.  The  student  will  find  that  if  he  tries  to  imagine  his  melodies  as  being  performed  in  some  definite 
"tempo"  he  will  find  it  easier  to  obtain  melodic  ideas. 

INVENTION. 

Compose  several  four-measure  "phrases"  in  various  meters,  in  various  keys  and  in  both  major 
and  minor  modes,  making  them  as  imlike  each  other  as  possible. 

These  melodies  should  not  all  begin  on  the  same  part  of  the  measure,  but  in  every  case,  the 
sum  of  the  measures  in  each  melody,  whether  fractional  measures  are  employed  or  not,  should 
be  e.xactlv  four. 


Chapter  EigJit. 

iiARMo.\iZATio>:  OF  Mi:f.or)ii-:s. 

\\  hile  it  is  not  aKva\s  the  case,  every  tone  of  a  melody,  particularly  a  simple  one,  may  be  accom- 
panied or  harmonized  by  a  chord  of  which  it  is  a  member. 
The  following  rules  of  procedure  are  to  be  observed: 

(1)  Write    out    the   melody,    placing    it    in  con\cnient  range  for  a  .Soprano  part. 

(2)  Over  each  note,  write  tiic  Roman  m'meral  corresponding  to  the  chord  with  which  the 
note  is  to  be  harmonized,  .\void  the  implication  of  clumsy  chord  succession.  Be  sure  that 
the  first  and  last  chords  of  each  exercise  are  chords  of  the  Tonic. 

(?i)  Write  the  Bass  next,  using  in  it  the  roots  of  the  chords  selected  :  i.  c.  provide  for  each 
ilionl  being  in  Root  position. 

(4)  Carefully  fill  in  the  middle  parts  of  the  first  chord,  providing  as  far  as  possible  for 
correct  spacing  of  the  succeeding  chords.  If.  for  instance,  the  melody  steadily  rises,  the  .Mto 
of  the  first  Tunic  chord  is  best  written  as  clo>cI\-  as  i>ossible  to  the  First  note  of  the  Soprano. 
This  will  be  of  assistance  in  keeping  the  .\1tn  .ukI  Soprano  within  an  agreeable  distance  of  each 
other. 

(5)  I'ili  in  liie  remainder  of  the  middle  p.irts.  .\s  a  rule,  it  is  best  to  tie  the  common 
tone  first  when  filling  in  each  chord.  Attention  to  the  treatment  of  the  active  tones  will  gen- 
erally result  in  correct  doubling  and  s)):icin;;.  .illhough  certain  exceptions  often  arise.  These 
will  be  disciiscd  in  the  next  section. 

'i'lie   following  rules  will  naturally  arise: 

(1)    1  l;uini>ni/c    the    tirsl.    lliird,    and    fifth  degrees  of  the  scale  with  the  Tonic  chord. 


16 


Chapter  Mne 


(2)  Harmonize  the  fourth  and  sixth  degrees  of  the  scale  with  the  Subdominant  chord. 

(3)  Harmonize  the  second  and  seventh  degrees  of  the  scale  with  the  Dominant  chord. 

(4)  The  first  degree  of  the  scale  may  be  harmonized  with  the  Subdominant,  except  at  the 
first  or  last  notes  of  the  exercise,  and  the  fifth  degree  of  the  scale  may  be  harmonized  with  the 
Dominant,  but  neither  of  these  harmonizations  may  be  used  if  it  gives  rise  to  the  forbidden  pro- 
gression of  Dominant  to  Subdominant. 

(5)  An  active  tone  may  be  led  in  a  direction  contrary  to  its  tendency  either  conjunctly  or 
by  a  skip  of  a  third  if  approached  conjunctly  in  this  contrary  direction. 

(6)  When  necessary,  a  principal  Triad  in  Root  position  may  occur  with  its  fifth  omitted  and 
Root  tripled. 

These  rules  apply  to  major  and  minor  modes  alike. 

Chord  repetition  suspends  the  rules  for  the  treatment  of  active  tones  during  the  repetition. 
c.  The  following  is  an  example  of  a  harmonized  melody: 


a_ 


4- 


5E^ 


^^ 


i    iOJ. 


§fS 


^^^ 


i 


■SI- 


1 


i       i 


^ 


J- 


£ 


J. 


KXERCISE. 

Harmonize    four   "phrases"    in   major    and  four  in  minor.     These  should  be  taken  from  the 
work  of  the  preceding  chapter  if  possible. 


Chapter  Nine. 

CHORD  INVERSION THE  CHORD  OF  THE  SIXTH. 

a.  When  the  third  of  a  triad  is  in  the  Bass,  the  fifth  and  root  of  the  triad  will  occur  above  it  at  the  dis- 
tances of  a  third  and  a  sixth  respectively.  The  fact  that  these  intervals  often  occur  plus  one  or 
more  octaves  makes  no  difference  in  the  general  effect  of  the  chord. 

h  Such  an  arrangement  of  the  chord  is  called  its  "first  inversion."  The  intervals  of  a  third  and  sixth 
above  the  Bass  cause  the  chord  to  be  also  known  by  those  intervals  as  its  name.  It  is  gen- 
erally called,  however,  simply  "the  chord  of  the  sixth."     The  following  is  an  illu.stration : 


%0 


A    J  .  j   i 


g 


=f=^ 


i  ^^ 


^•-  i^    r   I' 


^ 


Iaa 


^^ 


Chapter  JVine 


17 


c.  In  the  above  illustration,  observe  that  the  root  is  sometimes  doubled  and  sometimes  the  fifth.    This 

is  done  because  these  tones  are  the  most  important  ones  in  the  chord,  as  may  be  seen  in  the 
frequency  of  their  occurrence  in  the  tables  of  overtones  and  undertones.  At  present,  the  Bass 
of  a  principal  chord  of  the  sixth  should  never  be  doubled. 

EXERCISE. 

Build  up,  in  four  parts,  five  different  arrangements  of  the  first  inversion  of  each  of  the  pri- 
mary triads,  in  si.x  diflferent  keys.  Space  the  chords  as  before.  Write  an  Arabic  six  (6)  under 
the  Bass  of  each  chord  and  retain  the  custom  throughout  the  course. 

F..\U    TRAINING. 

Re  able  to  distinguish  Chords  of  the  Sixth  from  root  position  chords. 

d.  On  account  of  the  more  fluid  nature  of  the  Chord  of  the  Sixth,  a  skip  of  over  a  third  from  its 

Bass  should  be  avoided,  particularly  at  present,  since  its  Bass  is  neither  a  chord  root  or  a  prin- 
cipal tone  of  the  key.  This  rule  may  be  suspended  in  chord  repetition,  but.  as  a  rule,  it  is  better 
to  have  a  chord  inversion  follow  its  root  position  rather  than  precede  it.  .\  skip  to  a  chord  of 
the  Sixth  is  always  good  when  the  Bass  of  the  Chord  of  the  Sixth  can  be  led  in  a  direction 
contrary  to  that  of  the  skip. 

INVENTION. 

Invent  four  measure  chord  sequences  in  various  major  and  minor  keys  and  various  meters 
making  use  of  Chords  of  the  Sixth  to  secure  as  smooth  a  Bass  as  possible.  Figure  every  Six- 
Three.     Begin  and  end  with  the  Tonic  chord  in  root  position. 

e.  In  the  manipulation  of  the  Subdominant  Chord  of  the  Sixth,  if  the  root  of  the  chord  is  doubled,  one 

of  the  tones  should  be  treated  strictly,  as  an  active  tone,  and  the  other  may  be  treated  freely, 
as  a  chord  root:  /.  c,  allowed  to  move  freely;  if  the  fifth  of  the  chord  is  doubled,  the  two  tones 
should  not  be  treated  alike.  The  choice  of  these  doublings  and  treatments  should  be  governed 
by  the  doubling  and  spacing  of  the  following  chord  as  well  as  by  the  following  rule. 

/.  If  any  tw^o  parts  hold  the  interval  of  a  fifth,  they  should  not  move  into  another  fifth.     A  repeated 
fifth,  being  merely  a  prolongation  of  the  same  two  tones,  is  not  subject  to  this  rule. 
The  same  rule  applies  with  octaves. 


(J.  Some  progressions  of  the  Subdominant  Chord  of  the  Sixth  are  illustrated  below. 
that  the  rules  given  in  Chapter  I'our  are  still  applied  in  a  general  way. 


It  will  be  seen 


^.li    "^ 


J=^ 


IT   -Z 


^4^ 


^  J 


^^ 


a 


p^ 


i=^ 


6  6 

EXERCISE. 

Re-harmonize  two  of  the  inventions  used  in  the  last  chapter  in  major  keys,  and  two  in  minor 
keys,  applying  Chords  of  the  Sixth  to  secure  as  much  smoothness  as  possible  in  the  leading  of 
the  Bass.  Compose  two  new  phrases  in  major  keys  and  two  in  minor  ke)s.  and  harmonize  them 
ill  a   similar  manner,  applying  the  directions  given  in  the  preceding  chapter. 


18  Chapter  Ten 

On  account  of  llic  sniallness  of  the  liarnioiiic  vocalnilaiy  at  our  present  disposal,  it  will  be 
frequently  necessary  to  write  very  sustained  Alto  and  Tenor  parts.  This  deficiency  will  be 
readily  avoided  after  more  chords  are  understood. 


Chapter  Ten. 

CHORD   INVERSION CHORD  OF   THE   SIX-FOUR. 

a.  When  the  fifth  of  a  triad  is  in  the  Bass,  the  third  and  root  of  the  triad  will  occur  above  it  at  the 

distances  of  a   sixth  and   fourth   respectively.     (See  a  of  preceding  chapter.) 

b.  Such  an  arrangement  of  the  chord  is  called  its  "second  inversion."     The  intervals  of  a  sixth  and 

fourth  above  the  Bass  cause  the  chord  to  be  also  known  by  those  intervals  as  its  name,  and 
as  is  generally  called  the  "Chord  of  the  Sixth  and  Fourth,"  or,  more  familiarly,  the  "Six-four 
Chord."    The  following  is  an  illustration  : 


-a 0 — S-^ 

y n rr— 

1    9      ^ 

1     ''         0          =1 

"^ Q 

n n O 

U —jf, ^            ^f. 

""—^ ^-     ^ 

c.  Observe  that  the  fifth,  which  is  in  the  Bass,  is  always  doubled.     This  is  an  invariable  rule  in  ele- 

mentary composition. 

Compare  this  section  with  Section  c  of  the  preceding  chapter. 

EXERCISK. 

Build  up,  in  four  parLs,  live  dilfcrciU  arraiigcnicnts  of  the  second  inversion  of  each  of  the 
primary  triads,  in  six  different  keys.  Space  the  chords  as  before.  Write  an  Arabic  six  under 
the  Bass  of  each  chord,  under  that  an  Arabic  four,  and  retain  this  custom  throughout  the  course. 

i:.\R    TR.MNING. 

Be  able  to  distinguish  Chords  of  the  Sixth  and  Fourth  from  Chords  of  the  Sixth  and  from 
root  position  chords. 

d.  On  account  of  the  nature  of  the  Six-four  Chord,  wiiich  will  Ije  understood  by  further  experience, 

the  following  rules  for  its  use  will  be  found  necessary: 

( 1 )  The  Bass  of  a  Six-four  Chord  may  be  preceded  or  followed  by  the  same  scale  degree. 
This  includes  a  skip  of  an  octave  in  the  Bass  part. 

(2)  The  Bass  of  a  Six-four  Chord  may  be  preceded  or  followed  by  a  tone  one  degree 
above  or  below  it. 

(3)  The  Bass  of  a  Six-four  Chord,  if  it  docs  not  skip  an  octave  as  in  d),  may  skip  onlv  to 
the  root  or  third  of  the  same  chord,  although  it  is  better  to  have  a  root  position  precede  an  inver- 
sion of  the  same  chord  rather  than  follow  it. 

INVENTION. 

Invent  various  chord  sequences  of  four  measures,  each  in  different  keys  and  meters,  making 


Chapter  Eleven 


19 


use  of  Six-four  Chords  to  secure  as  smooth  a  Bass  as  possible.    Figure  each  inversion  accord- 
ing to  rule.     Begin  and  end  as  before. 

c.  What  was  said  in  Section  e  of  the  preceding  chapter  applies  generally  here. 

Section  /  of  the  preceding  chapter  applies  here  and  continues  to  do  so  until  further  notice. 
/.  The  use  of  Six-four  Chords  permits  the  occasional  tying  of  Bass  notes.     Unless  the  rule  for  the 
correct  treatment  of  Six-four  Chords  should  be  thereby  broken,  it  is  better  to  avoid  the  tying 
of  an  unaccented  tone  in  the  Bass  to  an  accented  tone. 

This  rule  applies  to  all  the  voices,  but  the  student  should  not  carry  it  out  too  strictly  until 
it  can  be  observed  with  but  little  trouble. 

g.  The  following  is  an  example  of  harmonization  by  means  of  the  use  of  root  position  chords  and  both 
forms  of  inversion.    Observe  the  further  gain  in  smoothness  in  the  treatment  of  the  Bass. 


^ 


^ 


^m 


■e- 


A 


t 


f 


:^ 


^ 


-^== 


■Ai  i 


1 


4:- 


EXERCISE.  \-  \. 

Re-harmonize  two  of  the  inventions  used  in  the  last  chapter  in  major  keys,  and  two  in  minor 
keys,  applying  both  forms  of  inversion  to  secure  as  much  smoothness  as  possible  in  the  leading 
of  the  Bass.  Compose  two  new  phrases  in  major  keys,  and  two  in  minor  keys,  and  harmonize 
them  in  a  similar  manner,  applying  the  directions  given  in  Chapter  Eight. 


Chapter  Eleven. 

THE  MUSICAL  SENTENCE  OR   "PERIOD." 
THE  SIMPLER  CADENCES    AND    THEIR    USE. 

((.  1  he  simplcbl  type  of  Period  is  composed  of  two  |)hrases.  I'or  the  sake  of  completeness,  each  Period 
should  end  with  tlio  Tonic  chord  in  root  position,  hut  tliis  cannot  be  done  at  the  end  of  the  first 
l>hrase,  as  the  ending  would  thereby  be  anticipated. 

/).  The  harmonization  of  the  ends  of  the  phrases  is  of  the  greatest  importance.  To  completely  end  the 
Period,  wc  must  close  with  the  Tonic  chord  in  root  position;  to  avoid  ending  too  soon  we  must 
end  the  first  phrase  with  either  the  I\'^  or  V,  the  latter  being  generally  to  be  preferred.  It  is 
occasionally  necessary  to  end  the  first  Phrase  with  the  Tonic  chord,  but  in  this  case  it  should 
occur  in  first  inversion  only. 

c.  Each  phrase  should  end  with  two  different  chords.    These  sets  of  two  chords  are  called  "Cadences." 
When  a  Cadence  is  made  of  the  I  and  Y  it  is  called  an  .\uthentic  Cadence ;  when  it  is  made 
of  the  I  and  the  I\^  it  is  called  a  Plagal  Cadence;  when  both  the  I\'  and  the  \'  are  concerned, 
it  is  called  a  Mixed  Cadence. 


20 


Chapter  Twelve 


When  the  I  is  the  last  chord  in  tlie  Cadence,  tlie  Cadence  is  said  to  be  a  "Full  Cadence;" 
when  the  I  precedes  the  IV  or  the  V,  we  obtain  a  "Half  Cadence." 

When  both  chords  are  in  root  position  we  obtain  a  Perfect  Cadence ;  when  either  or  both  of 
the  chords  is  inverted,  we  obtain  an  "Imperfect  Cadence." 

(/.  At  least  one  of  the  chords  of  a  Cadence  should  occur  on  an  accented  beat ;  when  greater  strength  is 
desired,  the  melody  may  be  so  arranged  that  both  cliords  may  occupy  accented  beats. 

e.  Each  Period  should  end  with  a  full  cadence,  either  "Authentic"  or  "Plagal,"  "Perfect"  or  "Imper- 
fect," although  the  perfect  form  is  to  be  preferred  here. 

The  first  phrase  should  end  with  a  "Half  Cadence"  or  rarely,  with  an  "Imperfect  Full 
Cadence."    A  very  useful  form  is  lo-V,  with  the  I  on  the  accented  beat. 

/.  A  six-four  chord  should  never  be  used  as  a  second  chord  in  a  Cadence. 

g.  We  may  extend  the  final  Cadence  by  using  the  "Mixed"  form  IV-V-I,  or  IV-I-V-I. 

h.  The  study  of  the  more  complicated  Cadences  and  their  uses  will  be  left  for  a  later  stage. 

j.  The  following  is  an  example  of  the  construction  of  a  Period  and  the  treatment  of  the  Cadences.  It 
will  be  observed  that  the  beginning  of  the  two  Phrases  are  alike.  This  is  one  of  the  simplest 
ways  of  obtaining  unity  in  the  Period  Form.    The  student  should  often  make  use  of  this  device. 


INVENTION. 

Take  four  Phrases  in  major  keys,  and  four  in  minor  keys,  from  the  inventions  of  Chapter 
Seven.    Expand  them  into  Periods,  harmonizing  them  with  due  regard  to  the  necessary  Cadences. 


Chapter  Twelve. 

THE    SECONDARY    TRIADS    1  .\    TIIIC   MAJOR    MODK. 

«.  The  three  chords  that  we  know  in  the  major  mode  are  shown  in   Figure  24. 


c'^.*^ 


I? 


4= 


We  find,  however,  that  it  is  possible  to  erect  and  use  a  Triad  on  every  degree  of  the  scale, 
as  shown  in  Figure  23. 


^^^^ 


Chapter  Twelve 


21 


b.  As  none  of  these  new  chords  are  made  from  the  overtones  of  their  roots,  we  must  find  a  way  of  ac- 

counting for  them,  different  from  that  employed  for  the  I,  IV,  and  Y. 

We  find  that  the  II,  III,  and  \'I  are  minor  triads,  and  as  we  know  that  minor  triads 
may  be  formed  by  the  inferior  resonance  of  their  fifths,  we  will  investigate  the  origin  of  these 
chords,  leaving  the  \\\.  which  is  a  diminished  chord,  to  be  studied  later  in  Chapter  Nineteen. 

c.  We  find  that  we  may  account  for  the  II,  III,  and  \'I  as  undertone  chords  of  the  sixth,  seventh  and 

third  degrees  of  the  scale,  as  shown  in  Figure  26. 


d.  The  relation  of  these  new  chords  to  the  key  center  is  shown  by  means  of  Figure  27. 


1 


^ 


M 


.  a     g; 


It  will  be  seen  that  they  are  related  to  the  key  center,  not  directly,  as  are  the  primary  Triads, 
but  indirectly,  by  being  undertone  chords  of  the  thirds  of  the  principal  Triads. 

The  impression  they  make  upon  the  ear  agrees  with  this  observation  as  they  do  not  determine 
the  key  as  clearly  as  do  the  primary  Triads.  I-'or  this  reason  they  are  called  "Secondary 
Triads." 

c.  Each  secondary  Triad  will  be  found  to  have  two  notes  in  common  with  the  principal  Triad  on  the 
Third  of  which  it  depends.  This  fact  causes  us  to  call  such  a  chord  the  "Secondary"  of  the 
Triad  on  which  it  depends,  and  also  to  call  the  primary  chord  the  "Primary"  of  its  dependent 
Triail. 

/.  i-rom  the  tact  that  cacli  ijrimars'  and  its  secondary  conlain  common  tones  wliich  are  noticed  as  l)eing 
characteristic  of  each  primary,  such  as  the  possession  of  the  leading  tone  by  both  the  III  and  the 
\'.  we  concludi-  that  each  secondary  belongs  to  the  same  "family"  as  its  primary. 

g.  The  idea  of  "chord  families"  is  most  imiiortant  and  useful  in  our  study  of  harmony.  By  accounting 
for  any  chord  as  a  member  of  either  the  Tonic,  Dominant,  or  Subdominant  families,  we  are  able 
to  greatly  simplify  our  work  as  well  as  the  bet.er  to  classify  our  vocabulary  of  chords. 

//.  .\  secondary  Triad  should  follow,  and  not  precede,  its  own  primary,  but  may  be  jjreceded  by  any 
other  primary  triad. 

.\  secondary  Triad  may  proceed  to  another  secondary  Triad,  except  in  the  cases  of  the  pro- 
gressions II-llI  and  111-11.  which  suggest  the  relative  minor  key  too  strongly  to  be  of  nuich  use 
at  present. 


22  Chapter  Tliirteen 

i.  A  secondary  Triad  in  root  position  may  double  its  root  like  a  principal  Triad.  But  as  its  third  is  a 
principal  tone  of  the  scale,  that  member  may  be  doubled,  as  well.  Good  voice  leading  should 
govern  the  choice. 

In  its  first  inversion,  it  may  double  its  root  as  before,  or  its  fifth  like  a  principal  Triad,  or  its 
third  for  the  reason  just  given. 

The  second  inversion  of  secondary  Triads  should  be  doubled  exactly  the  same  as  the  second 
inversion  of  principal  Triads. 

j.  The  Bass  of  the  root  position  and  the  two  inversions  should  be  figured  and  treated  the  same  as  the 
Bass  of  root  position  and  similar  inversions  of  the  principal  Triads. 

k.  Active  tones  should  be  treated  as  before. 

When  an  active  tone  is  taken  as  a  root  it  may  progress  freely. 

INVENTION. 

Invent  four  chord  sequences  eight  measures  in  length  in  various  meters  and  various  major 
keys,  making  frequent  use  of  secondary  Triads  in  their  root  positions  and  their  inversions.  Be- 
gin with  the  Tonic  chord  in  root  position,  and  close  with  a  "Perfect  Full  Cadence." 

/.  Secondary  Triads  have  not  enough  harmonic  strength  to  be  of  much  value  in  Cadences  at  present. 


Chapter  Thirteen. 

H.■\RMONIZ.^TION     OF    MELODIES,    USING    SIXONDARV    TRIADS MAJOR     MODE. 

a.  The  methods  indicated  in  Chapter  Eight  may  be  extended  to  include  the  secondary  Triads  of  the 
major  mode  as  additional  resources. 

/'.  The  use  of  secondary  Triads  imparts  variety  and  gives  new  opportunities  for  wider  choices  in  voice 
leading. 

All  the  parts  should  be  led  as  melodiously  as  possible.  While  occasional  skips  are  good,  if 
followed  by  conjunct  motion  in  a  direction  contrary  to  the  skip,  frequent  skips  should  be  avoided, 
as  they  generally  produce  a  disjointed  effect. 

f.  The  Bass  of  the  first  inversion  of  a  secondary  Triad  is  allowed  to  skip  over  a  third  on  account  of  its 
importance  as  a  principal  tone  of  the  key. 

il.  The  second  inversion  of  a  secondary  Triad  .should  not  be  used  on  a  strongly  accented  beat  unless  all 
the  voices  move  very  smoothly,  as  otherwise  such  an  inversion  frequently  indicates  a  modula- 
tion,— a  resource  which  is  at  present  beyond  our  reach. 

c.  -An  exception  in  the  treatment  of  six-four  chords  is  allowed  in  the  case  of  the  Tonic  Six-four  by 
preceding  this  chord  by  the  II  in  root  position.  The  use  of  this  exception  furnishes  us  with  a 
valuable  "Semi-cadence"  in  the  form  II-I»-\'.  We  mav  extend  this  to  a  "Full  Cadence"  bv  fol- 
lowing  the  V  by  the  I. 

EXERCISE. 

Re-harmonize  several  Periods  in  the  major  mode  written  in  the  work  of  Chapter  Eleven, 
making  frequent  use  of  the  secondary  Triads  and  their  inversions. 

New  melodies  in  the  major  mode  should  be  also  written  with  the  effect  of  the  secondary 
chords  in  mind  and  harmonized  with  the  use  of  these  chords. 

The   beginnings   and   Cadences   in   all    this  work  should  be  treated  as  before. 


Chapter  Fourteen  23 

EAR    TRAINING. 

Be  able  to  distinguish  the  secondary  Triads  from  the  primary  Triads  by  their  sound  alone, 
when  listening  to  such  work  as  has  just  been  done. 

This  work  should  be  extended  to  include  tlie  ability  to  distinguish  the  different  inversions 
of  the  secondary  Triads  from  the  different  inversions  of  the  primary  Triads,  both  by  chord  name 
and  chord  inversion. 


Chapter  Fourteen. 

THE   SECONDARY   TRIAD.-^    IN    THE    MINOR    MODE. 
THEIR    EVOLUTION,    TREATMENT,     AND     APPLICATION. 

a.  The  three  primary  triads  of  the  minor  mode  are  shown  in  Figure  28. 


^•\*  12 


=^ 


bjilHlg     bo(H)g       '^       I 


We  find,  however,  that  it  is  possible  to  erect  and  use  a  triad  on  every  degree  of  the  scale  as 

shown  in   Figure  29. 


We  have  already  found  that  the  seventh  degree  of  the  minor  scale  may  lie  either  a  whole  step 
from  the  Tonic  or  a  half  step  from  it,  i.  e.,  in  the  given  figure  we  may  use  either  the  natural  of  the 
raised  seventh.  The  former  will  be  employed  as  more  useful  at  present  in  the  VII  of  the  minor 
mode :  we  have  found  either  form  of  the  seventh  degree  possible  in  the  \' ;  as  the  fifth  of  the 
III  is  the  seventh  degree  of  the  scale,  the  use  of  one  form  of  the  seventh  degree  will  give  us  a 
major  triad  for  the  III,  and  the  use  of  the  other  form  of  the  seventh  degree  will  change  this  into 
an  augmented  chord.    Both  forms  may  be  used. 

/'.  Upon  examination  wc  will  find  that  the  three  principal  triads  of  the  minor  mode  not  only  establish 
the  scale  of  the  mode,  but  are  the  only  undertone  chords  which  are  possible  in  diatonic  harmony, 
that  is.  harmony  made  from  the  notes  of  the  scale  only.  We  find  major  triads,  or  overtone 
cliurds,  existing  on  the  third,  sixth,  and  sevenlh  degrees  of  ihe  theoretical  (descending)  minor 
scale,  analogous  to  the  formation  by  undertones  of  the  secondary  triads  in  the  major  mode,  upon 
the  third,  sixth,  and  sevenlh  degrees  respectively.  Tlie  diminished  chords  will  he  studied  later, 
in  Chapter  Nineteen. 

f.  In  the  following  figure,  we  find  the  theoretical  minor  scale,  with  its  three  principal  triads  written  as 
undertone  chords  of  the  first,  fourth,  and  fifth  degrees. 

K^*.  So 


^^     "  ^"  bg    1.;]    ^o    l.„ 


24 


Chapter  Fourteen 

In  tlie  modern  arrangement  of  the  minor  scale,  the  true  relationships  are  maintained,  but  the 
\'II  and  VI  become  the  Yl  and  VII,  respectively,  just  as  was  the  case  of  the  IV  and  V.  Sec- 
ondary triads  are  foimd  on  the  third,  sixth,  and  seventh  degrees  of  the  scale.  This  is  illustrated 
in  the  upper  staff  of  Figure  31,  which  is  to  be  compared  with  I'igure  26  of  Chapter  Twelve.  The 
lower  staff  of  Figure  31  places  these  chords  as  they  are  found  in  the  modern  minor  scale. 


5z^  31 


ba 


'i     '  ^f   b§    Hl^ 


m 


fcfe 


08        0 


m 


d.  The  relation  of  these  chords  to  the  key  center  is  shown  in  Figure  32. 


-32-  1,.  V 


It  will  be  seen  that  they  are  related  to  the  key  center,  not  directly,  as  are  the  primary  triads, 
but  indirectly,  by  being  overtone  chords  of  the  thirds  of  the  principal  triads.  (Compare  with 
Section  d  of  Chapter  Twelve.) 
e.  By  the  same  methods  used  in  Sections  e,  f,  and  g  of  Chapter  Twelve,  we  can  classify  these  triads  ac- 
cording to  the  idea  of  "chord  families." 
/.  We  may  now  compare  the  classification  of  the  chords  that  we  know  in  the  major  and  minor  modes 
according  to  their  "families." 

In  the  major  mode,  the  Tonic  family  is  composed  of  tiie  I  and  the  VI,  the  Dominant  family 

is  composed  of  the  V  and  the  III,  the  Subdominant  family  of  the  IV  and  the  II. 

In  the  minor  mode,  the  Tonic  family  is  composed  of  the  I  and  the  III,  the  Dominant  family 
of  the  \'  and  the  \TI.  the  Subdominant  family  of  the  I\'  and  \T. 

r/.   I'litil  further  notice,  the  \'II  of  the  minor  key  is  to  be  invariably  used  as  a  major  triad. 

/(.  When  the  III  is  used  as  an  augmented  triad,  the  presence  of  the  leading  tone  gives  it  certain  Dom- 
inant characteristics.  In  this  form,  the  active  tone  tendency  of  the  leading  tone  should  be  care- 
fully observed. 

When  the  III  is  used  as  a  major  chord,  its  fifth  demands  no  more  than  a  smooth  melodious 
treatment. 

I.  Sections  h.  i.  ;'.  and  k  of  Chapter  Twelve  apply  here  also,  with  the  following  exceptions: 

The  progressions  II-III  and  III-II  are  impossiljle  at  present  as  we  have  not  yet  treated  of  the 
II :  the  progressions  VI-VII  and  VII-VI  should  be  avoided  as  they  suggest  the  relative  major 
kev  too  strongly  to  be  of  much  use  at  present.  (Compare  the  similar  progression  III-II  and 
II-lII  in  major.) 


Chapter  Fifteen  25 

The  thirds  of  the  \'I  and  VII  may  be  doubled  as  in  Section  i,  of  Chapter  Twelve,  but  as  they 
are  major  thirds  from  their  roots,  this  doubling  will  sound  faulty  unless  brought  in  by  smooth 
progression  to  the  notes  so  doubled. 

When  the  Til  is  taken  as  an  augmented  triad  it  may  be  led  to  the  I  without  bad  effect,  on 
account  of  the  tendency  of  the  leading  tone  to  progress  to  the  Tonic. 

When  the  III  is  taken  as  a  major  triad,  it  should  be  treated  in  manner  similar  to  the  VI  and 
\TI :  when  taken  as  an  augmented  triad,  its  fifth  should  never  be  doubled,  as  the  leading  tone  in 
the  minor  mode  becomes  over-prominent  whenever  it  is  doubled.  Consequently,  in  root  position 
and  first  inversion,  this  augmented  chord  should  double  either  its  root  or  third ;  in  its  second  in- 
version, which  is  rare,  it  should  generally  double  its  third. 

Although  neither  root  nor  third  of  the  \'II  are  principal  tones  of  the  scale,  the  chord  itself  is 
of  such  comparatively  rare  occurrence  that  any  inconsistency  in  regard  to  its  method  of  doubling 
may  be  safely  passed  over. 

INVE.VTION. 

Invent  four  chord  sequences,  eight  measures  in  length,  in  various  meters  and  various  minor 
keys,  making  frequent  use  of  the  secondary  triads  in  their  root  positions  and  their  inversions. 
Begin  and  end  as  before. 

;'.  The  material  of  Chapter  Thirteen  may  be  applied  to  the  harmonization  of  melodies  using  secondary 
triads  in  the  minor  mode  w'ith  the  following  exceptions : 

The  Bass  of  the  first  inversion  of  a  secondary  triad  in  the  minor  mode  should  not  be  treated 
quite  so  freely  as  the  Bass  of  a  similar  chord  in  the  major  mode.  This  is  particularly  applicable 
in  the  case  of  the  Bass  of  the  first  inversion  of  the  VII. 

Section  d  of  Chapter  Thirteen  should  be  noted  and  carefully  applied  here  also. 

As  we  have  not  yet  studied  the  II,  Section  e  of  Chapter  Thirteen  cannot  be  applied  as  yet. 

EXERCISE. 

Work  out  the  exercise  in  Chapter  Thirteen,  making  use  of  periods  in  the  minor  mode  as 
before,  as  well  as  in  new  keys. 

FAR    TRAINING. 

Work  out  the  ear  training  of  Chapter  Thirteen,  applying  it  to  the  various  triads  of  the  minor 
mode. 


Cli'ipfrr  Fifteen. 

lUli    .VPHl.KATION     r)r    TIIK    CONTRAPfNTAI.    DEVICKS    Ol-      "\l  XIMARY     NOTES."     "PASSING     NOTES,"     AND 
"CII-\NG1\(^.    notes"    to    THK    .MATEKIAI.    (i|     (  II  M'TKKS    THIRTEEN    A.VD    Kol'RTEEN. 

(  If  Strict  Counterpoint  has  not  been  studied  concurrently  with  this  course  in  harmony,  this 
chapter  should  be  omitted. ) 

(J.  If  the  student  has  been  studying  Counterpoint  along  with  this  course  in  harmony,  he  will  be  familiar 
with  the  devices  mentioned  in  the  title  of  this  chapter.  The  following  directions  will  assist  him 
in  applying  these  devices  to  his  studies  in  harmony. 

b.  The  principal  uses  of  these  devices  should  be  found  at  present  in  the  evolution  of  melody,  so  their 
prcsi'ut  api)lication  should  generally  occur  as  a  means  for  obtaining  more  melody  in  the  Soprano. 
Tlu-ir  .ipplication  to  the  Bass  part  will  be  .ilways  valuable,  as  the  Bass  is  thereby  rendered  more 


25  Chapter  Fifteen 

flowing.  The  use  of  sucli  devices  in  both  Soprano  and  Bass  at  once  presents  a  problem  that 
should  be  avoided  at  present,  as  its  correct  solution  demands  considerable  maturity  of  contra- 
puntal feeling.  .\n  occasional  passing  or  auxiliary  note  may  be  u.sed  in  the  inner  parts  if  its  effect 
is  good,  but  here,  again,  the  problem  is  too  difficult  to  admit  of  extended  treatment  at  present. 

c.  As  a  rule,  none  of  these  devices  should  be  used  at  the  same  time  that  the  other  voices  are  sounding  a 

new  chord,  although  rare  exceptions  to  this  ruk-  may  be  permitted  if  their  eflfect  is  good. 

d.  These  devices  are  best  employed  in  quarter  notes  or  notes  of  smaller  value,  but  as  a  rule  nothing  is 

gained  at  present  by  the  use  of  notes  shorter   than  a   sixteenth. 

Dotted  notes  of  various  values  may  be  employed  provided  that  the  note  dotted  is  generally 
a  chord  tone. 

e.  As  these  devices  most  frequently  use  notes  foreign  to  the  chord  prevailing  at  the  time  of  their  use, 

they  are  classed  among  the  so-called  "non-harmonic"  devices. 

/.  When  employing  such  means  to  obtain  more  melody  in  the  Soprano,  care  should  be  taken  to  avoid 
rhythmic  monotony,  or  an  overloading  of  the  Soprano  with  non-harmonic  tones.  Advantage 
should  be  taken  of  opportunities  for  the  use  of  dotted  notes.  A  melodious  eflfect  is  as  frequently 
gained  by  the  avoidance  of  a  passing  note  as  it  is  by  its  use :  in  fact,  a  careful  cultivation  of 
taste  in  such  matters  is  the  only  possible  procedure. 

g.  On  the  other  hand,  the  attainment  of  a  freely  flowing  Bass  necessitates  the  maintenance  of  a  stead- 
ily moving  series  of  tones,  all  of  which,  except  the  last,  should  be  of  the  same  value,  or  as  near 
to  some  certain  value  as  possible.     This  will  require  a  copious  use  of  non-harmonious  devices. 

EXERCISE. 

Work  over  four  exercises  from  Chapters  Thirteen  and  Fourteen,  using  the  given  non- 
harmonic  devices  for  the  evolution  of  further  melodic  effect  in  the  Soprano  parts. 

Work  over  four  more  exercises  from  the  same  sources,  obtaining  smoothly  flowing  Basses. 
Make  the  Basses  in  two  of  the  exercises  move,  as  far  as  possible,  in  steadily  flowing  quarter 
notes,  and  the  Basses  of  the  other  two  exercises  as  far  as  possible  in  steadily  flowing  eighth 
notes. 

h.  These  latter  four  exercises  present  an  application  in  their  Bass  parts  of  what  may  be  studied  later 
under  the  title  of  Free  Counterpoint. 


Chapter  Sixteen 


27 


PART  II 

THE  DIATONIC  DISSONANCES. 

Chapter  Sixteen. 

THE    EVOLUTION    AXD    TREATMENT    OF    THE    PKIMARV   SRXKNTH    IN    MAJOR  AND   MINOR  KEYS. 

a.  With  the  exception  of  one  diminished  triad  in  the  major  mode  and  two  in  the  minor  mode,  the  pos- 
sible series  of  diatonic  triads,  or  chords  of  three  different  notes,  is  now  complete.  Leaving  the 
explanation  of  these  diminished  triads  to  Chapter  Nineteen,  we  will  investigate  the  following 
scries  of  overtones  in  the  effort  to  find  a  new  chord  which  necesarily  must  be  one  of  four  differ- 
ent notes. 


R 


■^.ii 


m 


T 


j    ^  ^M' 


b.  We  find  that  we  can  construct  such  a  chord  from  a  fundamental  and  its  first  six  overtones,  (first  to 

seventh  partials )  which  may  be  reduced  to  the  form  of  a  major  triad  plus  a  minor  seventh. 

c.  The  following  figure  shows  the  major  scale  with  this  chord  erected  on  each  of  its  degrees: 


fe^ 


^^^ 


It   will   he  sct-n  thai   tills  chord  may  occur  diatonically  on  the  fifth  degree  only. 

(/.  .As  this  chord  occurs  only  on  the  Dominant,  and  contains  the  interval  of  a  seventh,  the  chord  is 
calk-d  '"The  Chord  of  the   Doniinaul   .SeveiUli." 

(.  While  it  is  possible  to  find  an  undertone  seventh  chord  in  the  minor  mode,  corresponding  to  the 
dominant  seventh  in  the  major  key,  modern  practice  erects  and  treats  the  dominant  seventh  in 
the  minor  mode  as  it  does  in  the  major.  This  is  possible  when  the  raised  seventh  degree  is  used 
in  the  minor  scale. 

/'.  The  student  should  be  familiar  by  now  with  the  fact  that  all  dissonance  demands  resolution,  i.  e.,  a 
special  progression  according  to  the  nature  of  the  tones  forming  the  dissonant  interval  as  well 
as  to  the  interval  between  them,  in  the  root  position  of  the  dominant  seventh  chord  given  in 
Figure  i?>,  we  observe  the  dissonance  of  a  minor  seventii  between  the  root  and  seventh  and 
anotliiM-  dissonance  of  a  diminished  fifth  between  the  third  rmd  seventh.  This  throws  the  seventh 


28  Chapter  Sixteen 

of  tlie  chord  into  liigh  relief,  intensifying  its  active  tone  progression,  and  accounting  for  the  rule 
always  given  that  the  seventh  of  the  dominant  seventh  should  always  resolve  one  degree  down- 
ward. On  account  of  the  dissonance  between  the  seventh  and  the  third,  this  third,  which  is  the 
leading  tone  of  the  scale,  has  its  active  tone  tendancy  intensified  also,  and  consequently  gen- 
erally resolves  upward  to  the  first  degree  of  the  scale. 

g.  We  may  conclude,  then,  that  the  dominant  seventh  should  resolve  into  a  chord  containing  the  first 
and  third  degrees  of  the  scale.  The  chords  that  we  know  that  contain  both  of  these  scale 
degrees  are  the  I  and  VI,  and,  in  fact,  the  progressions  \',-I  and  Vj-VI  are  the  ones  most  gen- 
erally employed. 

h.  By  tying  over  the  leading  tone,  the  dominant  seventh  may  he  lead  to  the  III,  but  this  should  not  be 
often  done.  It  is  best,  at  present,  when  the  III  progresses  again  into  another  inversion  or 
arrangement  of  the  dominant  seventh. 

I.  The  several  possible  inversions  of  the  dominaiil  seventh  together  with  their  "figuring"  may  be 
studied  below.  It  will  be  seen  that  the  figures  indicate  the  distance  from  the  Bass  of  the  root 
and  seventh  of  the  chord,  the  other  intervals  being  understood. 

II o_        "S 


10      d"     ^ 


=f 


1  's  S  ^ 

j.  The  preceding  and  following  remarks  on  the  tre;itnient,  inversion,  and  figuring  of  the  dominant 
seventh  will  be  seen  to  apply  to  the  minor  as  well  as  to  the  major  mode. 

k.  In  the  treatment  of  this,  as  well  as  all  other  dissonant  chords,  it  is  generally  best  to  approach  the 
dissonance  by  oblique  or  contrary  motion. 

As  the  seventh  of  the  dominant  seventh  is  an  essential  part  of  the  chord,  it  need  not  be 
"prepared,"  as  suspended  dissonances  are  in  Strict  Counterpoint. 

/.  Any  chord  that  may  effectively  precede  the  V  may  precede  the  \'-.  Note  the  exception  in  Section  It 
of  this  chapter. 

Ml.  The  following  directions  apply  to  the  usual  proKiesions  noted  in  Section  g. 

In  resolvin5;  the  clnniinant  sc\cntli,  the  root  moves  cillicr  to  the  root  of  the  Tonic,  to  the  root 
of  the  VI,  or  is  sustained  as  a  common  tone,  the  third  of  the  dominant  seventh  rises  to  the  Tonic, 
the  fifth  of  the  dominant  seventh  falls  to  the  root  of  the  Tonic,  and  the  seventh  of  the  dominant 
seventh  resolves  to  the  third  of  the  Tonic.  These  movements  are  to  be  observed,  no  matter  in 
what  form  or  inversion  the  dominant  seventh  occurs. 

W^hen  the  dominant  seventh  occurs  in  root  position,  it  is  better  to  omit  the  fifth  and  double 
the  root,  except  when  harmonizing  the  second  degree  of  the  scale,  when  it  is  better  to  double 
the  root  and  omit  the  third,  but  this  rule  applies  to  the  progression  V;-I  only. 

11.  When  the  third  inversion  of  the  dominant  seventh  is  followed  by  the  first  inversion  of  the  Tonic,  the 
fifth  of  the  dominant  seventh  may  move  to  the  fifth  of  the  Tonic. 

o.  When  the  second  inversion  of  the  dominant  seventh  is  followed  by  the  first  inversion  of  the  Tonic, 
the  seventh  of  the  dominant  seventh  may  rise  to  the  fifth  of  the  Tonic,  on  account  of  the  note 
of  resolution  being  taken  by  the  Bass. 


Chapter  Seventeen 


29 


/>.  DcceiHive  cadence.  The  deceptive  cadence  arises  from  leading  the  dominant  seventli  to  the  triad 
on  the  sixth  degree,  instead  of  the  Tonic.  The  only  difference  between  these  two  chord  move- 
ments is  that  the  root  of  the  dominant  seventh  is  led  to  the  root  of  the  triad  on  the  si.xth  degree. 
The  dominant  seventh  should  always  occur  in  its  complete  form  in  the  deceptive  cadence. 

q.  In  the  major  mode  the  progression  X'.-VI  may  he  employed. 

r.  E.xampk'S  of  the  foregoing:     (Sees.  m.  )i.  o,  p,  q.) 


S 


Jikn,. 


/w 


O. 


t- 


AT 


je^A/. 


^ 


<^> 


*■^ 


E.\R    TR.MNING. 

Be  able  to  recognize  the  dominant  seventh  in  either  the  rot  position  or  any  inversion. 

EXERCISE. 

Write  the  dominant  seventh  in  root  position,  and  also  in  its  various  inversions,  in  various 
major  and  minor  keys,  preceding  each  dominr.nt  seventh  by  one  other  chord,  which  may  be  the 
dominnnt  when  desired,  and  following  it  by  a  chord  to  which  it  may  progress  correctly. 


Chupter  Seventeen. 

APPI.IC.STION    Ol-    CHAPTKR    .SIXTEEN     IN     II  ARMoN  IZl  NC    MEI.OniES   IX    MAJOR    AND   MINOR   MODES. 

a.  As  a  rule,  the  dominant  seventh  may  be  used  whenever  the  \'  would  be  possible,  excepting  in  the 
case  where  the  second  degree  of  the  scale  is  followed  in  the  given  melody  by  the  third.  If  the 
dominant  seventh  is  used  to  harmonize  the  second  degree  in  this  case,  faulty  doubling  in  the 
following  chord  will  result,  except  in  some  rare  cases  where  the  deceptive  cadence  would  be 
possible. 

/).  When,  in  the  given  melody,  the  fourth  degree  of  the  scale  is  followed  by  the  third,  the  dominant 
seventh  may  be  used  to  harmonize  the  fourth  degree. 

c.  The  progression  of  the  fifth  degree  to  the  sixth  in  the  given  melody  occasionally  gives  an  oppor- 
tuiiitv  fnr  ibe  dominant  seventh  to  harmonize  the  fifth  degree  and  make  a  deceptive  cadence. 


30 


Chapter  Eighteen 


d.  The  dominant  seventh  is  to  be  preferred  to  the  \'  in  a  final  cadence. 

e.  Opportunities  for  the  special  progressions  noted  in  Sections  o,  p,  q,  and  r  of  the  preceding  chapter 

should  be  carefully  noted  and  made  use  of. 
Example : 


i'lj''i:'f,r'f,l 


6     f     V 

EXERCISE. 

Re-harmonize  two  previously  written  periods  in  the  major  mode,  and  two  in  the  minor  mode, 
making  use  of  the  dominant  seventh  wherever  possible. 

INVENTION. 

Invent  four  new  periods  in  dit?erent  major  keys,  and  four  in  different  minor  keys,  using 
various  meters.  Invent  these  with  the  effect  of  the  dominant  seventh  in  mind.  Provide  occa- 
sional skips  in  the  melody  from  one  note  of  the  dominant  .seventh  to  another.  As  chord 
repetition  suspends  all  rult-s,  these  skips  may  be  always  handled  by  change  of  arrangement  or 
inversion  of  the  chord. 

Harmonize   these   melodies. 

Chapter  Eighteen. 

THI-:    CONTR.\PUNT.\L    niiVICK.S    OF    ".sr.^PKNSKlx"     .\XD    ".ANTICIP.VTIOX." 

(To  be  omitted  if  Chapter  Fifteen  has  not  been  studied.) 
o.  Various  suspensions,  already  in  use  by  the  pupil  in  his  exercises  in  Strict  Counterpoint,  may  be 
applied  to  his  exercises  in  harmony.    Their  treatment  is  the  same  in  both  cases,  subject  to  the 
following  restrictions : 

b.  Suspensions  should  always  occur  on  accented  beats,  generally  on  the  first  beat  of  the  measure :  their 

resolutions  should  generally  occur  on  a  less  strongly  accented  beat  of  the  measure. 

With  the  exception  of  the  suspension  of  the  ninth,  no  suspension  should  sound  at  the  same 
time  as  its  note  of  resolution. 

With  regard  to  the  suspensions  in  the  Bass,  we  may  state  that  the  suspension  to  the  third  of 
a  chord  is  the  only  one  whose  efTect  is  good  at  present. 

c.  Examples: 


Chapter  Js^neieen 


31 


d.  The  device  of  anticipation  may  be  defined  as  being  tlie  reverse  of  that  of  suspension,  seeing  that  it 

arises  from  a  note  anticipating  a  note  of  the  next  chord  instead  of  delaying  until  that  chord  is 
sounded  by  the  other  voices,  as  is  the  case  with  suspension. 

e.  Examples: 


f^.ii 


^ 


/'.  An  anticipation  should  occur  on  the  last  fraction  of  a  heat,  as  in  the  given  examples.  The  best 
anticipations  are  those  of  the  resolutions  of  the  leading  tone  and  of  the  seventh  of  the  domi- 
nant seventh,  although  others  are  by  no  means  forbidden. 

g.  Subject  to  the  above  restrictions,  suspensions  may  occur  in  any  voice,  but  anticipations  are  best 
employed  as  a  rule  in  the  Soprano  in  the  development  of  melody. 

f.KERCISE. 

Work   over   the   material   of  the   invention  in  the  preceding  chapter,  introducing  suspensions 
and  anticipations  whenever  their  effect  seems  good. 

Double  suspensions  may  be  employed  when  productive  of  good  effect. 


Chapter  J^ineteen. 

Tin-;  Kvoi.uTiox  .^wd  use  of  thic  TRi.\n  o.\  the  LE.\i)i.\f;  tone  in  m.\j()R  .vnd  minor  modes,  and  of 

THE    SUPERTONIC    TRIAD    IN    THE    MINOR    MODE. 

a.  On  accotuit  of  the  similarity  of  its  effect  and  trealnient,  the  triad  on  the  leading  tone  in  the  major 
and  minor  modes  is  to  be  regarded,  and  generally  treated,  as  the  dominant  seventh  chord  with  its 
root  omitted. 


* 


tl 


^ 


-&— 


-Q — 


In  the  al»ove  illustrations  it  will  be  seen  that  the    "root"    and    fifth    uf    this   chord    are    treated    in 
exactly  the  same  manner  as  the  third  and  seventh  of  the  dominant  seventh. 

When  the  fifth  of  the  chord  is  doubled,  one  of  these  tones  should  be  treated  like  the  seventh 
of  the  dominant  seventh,  while  the  other  should  be  led  upward  by  degree,  as  the  fact  of  its 
being  in  dissonance  with  its  apparent  root  forbids  it  to  skip. 


32 


Chaftter  Jfineteen 


The  third  of  tlie  chord  is  the  one  most  frequently  doubled.  As  an  active  tone,  it  tends  to 
resolve  to  the  first  or  third  degree  of  the  scale,  but  as  these  tones  are  taken  by  the  resolution 
of  the  apparent  root  and  the  fifth,  the  chord  third  may  be  treated  freely,  providing  the  following 
chord  is  correctly  doubled. 

c.  All  inversions  of  this  chord  are  possible,  hut  the   first   inversion   is   to   be   generally  preferred,  on 

account  of  the  weakness  produced  by  the  dissonances  of  the  diminished  fifth  or  augmented 
fourth  above  the  Bass  in  the  root  position  or  second  inversion.  .-Xs  this  chord  is  predominantly 
a  fragment  of  another  chord,  its  second  inversion  may  double  its  chord  fifth  or  chord  third. 

d.  The  chord  may  be  preceded  or  followed  in  the  same  manner  as  the  dominant  seventh,  although  the 

deceptive  cadence  in  the  minor  mode  is  sometimes  inconvenient. 


fC^.ffl 


I^dt 


b^     t>0      1? 


W 


b-e-    ^2. 


e.  The  above  figure  shows  the  theoretical  minor  scale  with  an  undertone  seventh  chord  projected  from 
its  fifth  degree.  If  the  generator  of  this  chord,  fifth  degree  of  the  scale,  be  omitted,  we  will 
have  a  chord  which  will  be  seen  to  be  the  supertonic  triad  in  the  harmonic  minor  mode.  This 
undertone  seventh  chord  will  be  discussed  more  fully  in  Chapter  Twenty-two. 

/.  As  the  root  and  fifth  of  the  supertonic  triad  in  minor  are  active  tones,  and  are  in  dissonance  to  each 
other  as  well,  their  active  tone  tendencies  become  more  accentuated.  Consequently  the  usual 
progression  of  this  chord  is  toward  chords  containing  the  third  and  fifth  of  the  scale,  although 
the  chord  may  also  move  to  dommant  chords  by  resolving  its  fifth  and  tying  over  its  apparent 
root.  Its  doublings  and  inversions  may  be  treated  in  the  same  way  as  those  of  any  other  sec- 
ondary triad,  although  as,  like  the  triad  on  the  leading  tone,  it  is  a  chord  fragment,  its  second 
inversion  may  double  either  fifth  or  third. 

(/.  Examples  of  treatment: 


The   progression    II-I«  noted   before   in   the  major  mode,  is  now  allowed  in  the  minor  mode. 
/(.  The  progressions  II-III   and  III-II  are  now  allowed  in  the  minor  mode,  but,  when  III  is  a  major 

triad,  will  somid  rather  too  much  like  progressions  in  the  relative  major. 


Chapter  Twenty 


33 


i;ak  training. 

Note  the  difference  in  effect  between  the  II  in  the  minor  mode  and  the  other  diminished 
chords  just  studied,  due  to  different  functions  in  tonality. 

EXERCISE. 

Re-harmonize   two   previously   written   periods  in  major  keys,  and  three  in  minor  keys,  using 
the  material  given   in  this  chapter  whenever  possible. 
/.  The  triad  on  the  leading  tone  has  not  sufficient  strength  for  a  cadence  chord. 


Chapter  Twenty. 

THE    EVOI.UTIOX    AND   TREATMENT    OF    THE    SECONDARY    SEVENTH    CHORDS    IN    THE    MAJOR    MODE. 

a.  In  Chapter  Sixteen,  we  found  the  dominant  seventh  to  be  the  only  diatonic  seventh  chord  made  by 
overtones.  Although  an  undertone  seventh  chord,  or  one  that  may  be  explained  as  such,  may  be 
noticed,  it  reallv  arises  from  a  different  source,  as  will  be  seen  later. 


f'i<g.^f■^ 


In  the  above  figure,  we  have  a  seventh  chord  erected  on  ever}'  degree  of  the  major  scale. 
That  erected  on  the  dominant  has  been  already  studied :  that  on  the  leading  tone,  just  alluded 
to,  will  be  discussed  in  Chapter  Twenty-eight. 

We  may  classify  the  remaining  seventh  chords  in  the  following  manner:  those  on  the  second, 
third,  and  sixth  degrees  may  be  said  to  consist  of  a  minor  triad  plus  a  minor  seventh ;  those  on 
the  first  and  fourth  degrees  may  be  said  to  consist  of  a  major  triad  plus  a  major  seventh.  The 
different  effect  of  these  two  classes  of  chords,  as  well  as  their  different  functions  in  music, 
leads  us  to  name  those  in  the  first  class  "Secondary  Sevenths"  and  those  in  the  second  class 
"Tertiary  Sevenths."     We  will  study  the  former  in  the  present  chapter. 

We  may  account  for  the  e.xistence  of  these  chords  by  regarding  them  both  as  combinations  of  triads 
and  as  approximations  of  the  natural  but  undiatonic  sevenths.     This  is  illustrated  below: 


f^.  H^H- 


TLI 


■%-] 


I 


5; 


|E 


■jr  "7 

This   tlieory  is  sustained   by  the    fact   that  tlie  characteristics  of  each  of  the-^e  chords  is  the 
sum  III'  ihc  characteristics  of  the  triads  of  which  it  is  com]iosed. 

(.  As  the  triads  composing  any  one  of  these  chords  belong  in  the  same  family,  we  may  say  that  the 
II;  belongs  to  tin-  snbdomitiant  family,  the  III-  to  the  dominant  family,  and  the  \'l,  to  the  tonic 
family. 

(/.  These  chords  may  be  preceded  or  followed  by  any  chord  which  would  make  a  good  progression 
with  either  of  their  constituents.      \ny  one  of  these  churck  may  be  preceded  by  either  of  its  con- 


34 


Chapter  Tiventy 


stituents,  but  no  constituent  should  follow  a  seventh  chord  of  which  it  is  a  member,  as  it  is  plain 

that  such  a  process  destroys  the  sense  of  progression, 
c.  When  a  secondary  seventh  chord  progresses  to  another  chord  which  does  not  contain  the  seventh  of 

the  seventh  chord  as  common  tone,  the  seventh  should  resolve  one  degree  downward  in  the  usual 

manner.    This  is  called  "Active  Resolution." 
Examples : 


i 


3C7 


^7 


^'s 


f.  When  the  seventh  of  the  seventh  chord  can  be  sustained  as  a  common  tone,  the  so-called  "Passive 

Resolution"  results. 
Examples : 


^ 


~5" 


•JEb. 


3C^ 


y.  Note  that,  in  active  resolution,  the  dissonances  resolve  by  the  downward  leading  of  the  seventh,  and 

that,  in  the  passive  resolution,  the  dissonances  resolve  by  the  progression  of  the  chord. 
//.  A  common  fault  is  illustrated  below : 


n            > 

^■^^ 

V        ^^ 

^ 

ym          Q 

0 

(w          B 

^^ 

^T 

r^-^ 

A- 

ti). 

/ 

w 

0 



3IT 

1 

F=" 

The  incorrect  resolution  of  a  seventh  to  an  octave  is  here  illustrated.     When  the  interval  of 
a  seventh  occurs,  conjunct  movement  of  the  lower  voice  is  against  the  nature  of  the  seventh.     It 
is  the  upper  voice  that  should  either  move  conjunctly  downward,  or  be  tied  over. 
J.  Active  tones  should  be  treated  as  before,  except  wliere  they  occur  as  the  roots  of  constituent  triads,  in 
which  case  they  may  progress  freely. 


Chapter  Twenty-one 


35 


L  These  chords  are  inverted  and  figured  the  same  way  as  tlie  dominant  seventh.     No  member  should 
ever  be  omitted  in  root  position  or  inversions,  as  some  of  the  quality  of  the  chord  will  be  lost. 

k.  In  this,  as  in  all  dissonant  chords,  the  dissonances  should  be  treated  as  directed  in  Chapter  Sixteen, 
Section  k. 

In  former  times,  the  sevenths  of  these  chords  were  always  "prepared"  like  suspensions;  this 
is  no  longer  necessary,  but  the  pupil  should  familiarize  himself  with  the  effect  of  the  prepared  as 
well  as  the  unprepared  seventh. 

JXERCISE. 

Write  the  secondary  sevenths  in  root  position  and  inversions  in  various  major  keys,  preceding 
each  chord  by  a  correctly  chosen  chord  and  resolving  it  either  actively  or  passively.     Obtain  as 
much  variety  in  the  choice  of  chords  as  possible. 
.  It  is  possible  to  lead  one  seventh  chord  to  another  as  shown  below. 


fA^.^^ 


f^.H€^- 


^ 


^1 


i^. 


5:7 


n.  The  progression  II, -Ij,  illustrated  at  the  end  of  the  above  example,  is  another  exception  allowed 
in  the  treatment  of  the  lo. 


Chapter  Txventy-one. 

APPLICATION    OF    CHAPTER    TWENTY    TO    HARMONIZATION  OF  MELODIES  IN  THE  MAJOR  MODE. 

The  following  method  will  be  found  useful : 

(1)  Select  a  melody  in  period-form,  and  indicate  its  cadences  by  writing  Roman  numbers 
over  the  melody  notes  at  such  places. 

(2)  After  filling  out  the  first  Tonic  chord,  look  carefully  over  the  melody  and  indicate  the 
use  of  secondary  sevenths  wherever  such  chords  seem  possible. 

While  it  is  true  that  any  note  of  the  scale  may  be  harmonized  with  the  secondary  seventh 
chords,  the  following  cases  may  l)e  eliminated  at  once: 

(A)  Where  the  conduct  of  the  melody  would  imi)ly  a  bad  chord  progression  in  case  a 
secondary  seventh  should  be  used. 

(B)  Where  the  chord  seventh  would  occur  in  the  Soprano  and  progress  upward  or  by 
skip  in  cither  direction.  This  may  be  sometimes  handled  when  the  melody  implies  chord 
repetition. 

(C)  Where  the  Soprano  would  not  hold  the  chord  seventh,  but  yet  would  make  a  skip  to 
some  note  not  in  the  same  seventh  chord. 


36 


Chapter  Twenty-two 


(D)  Where  more  than  three  seventh  chords  would  be  used  in  direct  succession.  (This  is 
often  unmusical.) 

(3)  The  three  lower  voices  may  now  be  added.  The  Bass  should  be  led  as  before,  but  it 
will  be  better  now  to  write  the  Alto  and  Tenor  with  it,  as  the  best  way  of  avoiding  useless 
difficulties.  Careful  attention  to  the  correct  treatment  of  the  seventh  chords  will  sometimes 
demand  a  change  in  the  previously  determined  scheme. 

EXERCISE. 

Apply  the  above  method  in  re-harmonizing  two  previously  written  periods  in  the  major 
mode. 

INVENTION. 

Write  four  new  periods  in  major  keys  with  the  effect  of  the  secondary  seventh  chords  in  mind. 
If  a  good  Bass  may  be  obtained,  melody  and  harmony  may  be  written  together.  The  pupil 
should  attempt  to  do  this  in  at  least  two  of  his  inventions  for  each  of  the  following  chapters. 

E.\R    TR.MNING. 

From  the  effect  of  the  seventh  chords  in  the  above  work,  learn  to  recognize  them  by  name, 
method  of  resolution,  and  inversion. 


Chapter  Tiventy-Uro. 

THE    EVOLUTION'    .AND    TRE.XTMENT    OF    THE    SECOXDARV    SF.XE.NTH     CHORDS    IN'    THE    MINOR    MODE. 

a.  In  this  chapter,  we  may  employ  the  same  general  method  used  in  Chapter  Twenty. 


-E-2- 


\0       u-^ 


A 


I 


!l 


p^^TJ^S   ^f    '-^     ^^     ^t    ^^    ^^     ^^o     ^^    ^^^       I 


(S) 


cS) 


@) 


In  the  above  figure,  we  have  a  seventh  chord  erected  on  every  degree  of  the  minor  scale. 
The  "melodic"  form  of  the  minor  scale  will  not  he  treated  for  the  present.  On  account  of  the 
two  forms  of  the  seventh  degree,  we  find  two  kinds  of  seventh  chords  on  the  first,  third,  fifth, 
and  seventh  degrees  of  the  scale.  For  convenience,  we  will  designate  all  seventh  chords  con- 
taining the  natural  seventh  degree  by  drawing  a  ring  around  the  Roman  number  that  indicates 
them.  For  instance,  in  the  above  figure  the  chord  C-E  flat-G-B  flat  will  be  indicated  by  (I,)  ;  the 
chord  C-E  flat-G-B  natural  will  be  indicated  by  I;,  and  so  on. 

For  reasons  given  in  the  third  paragraph  of  Section  a  of  Chapter  Twenty,  we  may  class- 
ify the  Ij,  the  (Til;),  and  III;,  and  \T.  as  "Tertiary  Sevenths,"  and  reserve  their  study 
for  a  later  time.  The  VTI;  will  be  discussed  in  Chapter  Twenty-eight;  the  V,  is  already  famil- 
iar as  the  dominant  seventh;  the  (VTI;)  is  identical  in  form  and  effect  with  the  dominant  sev- 
enth of  the  relative  major  key,  and  should  be  used  at  present  only  in  modulation  to  that  key,  or 
to  its  relative  minor.     This  leaves  us  for  genuine    secondary   sevenths   the    (I;),    II„   IV^   and 

(V;). 


Chapter  Tiventy-Two 


37 


b.  With  the  exception  of  the  II7,  these  secondary  sevenths  may  be  accounted  for,  classified,  and  treated 
like  the  secondary  seventlis  in  the  major  key,  so  we  will  find  that  the  {l^)  belongs  to  the  Tonic 
family,  the  IV,  to  the  snb-dominant  family,  and  the  (V,)  to  the  Dominant  family.  The  follow- 
ing illustration  will  make  this  clear: 


^^.  50 


^ 


^9. 

i8= 


1^^^  ^H    \%^\\  H  t^H~^ 


© 


IZ"? 


& 


c.  With  regard  to  the  11,,  the  following  theory  should  be  employed : 


w. 


^ 


L=^ 


The  above  figure  gives  an  undertone  series  extended  further  than  hitherto  used.  (It  should 
be  compared  with  the  overtone  series  treated  of  in  Chapter  Sixteen.)  From  the  generator  and 
the  first  six  undertones,  we  may  derive  the  chord  included  in  the  above  figure.  It  may  be 
defined,  as  we  think  it  upward,  as  a  diminished  triad  plus  a  minor  seventh,  but,  if  read  down- 
ward, will  be  seen  to  be  projected  downward  by  the  same  series  of  intervals  as  the  dominant 
seventh  is  built  upward.  This  chord  may  be  used  as  a  diatonic  undertone  chord  on  the  fifth 
degree  of  the  theoretical  minor  scale  only,  as  shown  below. 


A^.  S% 


i 


^=^ 


This  chord  reappears  in  the  modern  minor  mode  as  the  II7.  Its  origin  accounts  for  its  great 
strength.  The  student  has  i)robably  noticed  before  this  that  Dominant  harmony  in  the  theoret- 
ical minor  key  becomes  Subdominant  harmony  in  the  modern  minor  key,  so  that  it  should  cause 
no  confusion  if  we  classify  this  chord  as  the  strongest  seventh  chord  in  the  Subdominant  family. 

(/.  For  convenience,  we  may  also  take  the  view  of  this  chord  as  being  composed  of  the  II  and  IV  as 
constituents.  With  this  in  mind,  we  may  say  that  the  rules  for  its  introduction  and  progression 
are  the  same  as  for  all  other  secondary  seventh  chords,  excepting  only  that,  as  it  belongs  to  the 
same  family  as  the  1\',,  but  is  stronger  than  it,  it  may  follow  the  I\';,  but  should  not  pre- 
cede it. 

e.  Sections  /  and  m  of  Cha|)tor  Twenty  may  be  applied  here  also. 


^7  fS-f 


38 
Examples : 


EXERCISE. 


^O'.  S'd 


^ 


^■e-" 


® 


2=0= 


b-e- 


^ 


b-e- 


» 


2 

3C 


ffi 


b-^' 


b^ 


ra: 


jTv-j,       12^7 


in 


17 


Chapter  Twenty-four 


I6„ 


Work  out  the  exercise  of  Chapter  Twenty  in  various  minor  keys,  following  the  given  direc- 
tions. 
/.  When  using  the  minor  mode,  avoid  having  the  natural  seventh  of  the  scale  in  one  chord  and  the 
leading  tone  in  the  following  chord,  and  vice  versa. 


Chapter  Twenty-three. 

APPLICXTION    OF   CHAPTER   TWENTY-TWO   IN    HARMONIZATION    OF    MELODIES    IN    THE    MINOR    MODE. 

a.  The  method  given  in  Chapter  Twenty-one  may  Le  applied  in  the  minor  mode. 

b.  Careful  attention  should  be  directed  to  Division  2  of  Section  a,  as  well  as  to  Section  f  of  the  pre- 

ceding chapter. 

c.  The  exercise,  invention,  and  ear  training  of  Chapter  Twenty-one  should  now  be  worked  out  in  vari- 

ous keys,  but  in  the  minor  mode. 


Chapter  Twenty-four. 

THE  EVOLUTION,  TREATMENT,  AND  APPLICATION  OF  THE  TERTIARY  SEVENTHS  IN  THE   MAJOR   MODE. 

(J.  Returning  to  Figure  43  in  Chapter  Twenty,  we  find  tertiary  sevenths  on  the  first  and  fourth  degrees 
of  the  major  scale. 

We  may  apply  the  theory  of  origin  given  in  Chapter  Twenty  to  these  chords  as  well.     The 
application  of  this  theory  is  illustrated  below. 


^ 


i 


17 


12^7 


Note   that   tertiary   sevenths   are  composed  of  a  union  of  two  triads  not  in  the  same  family,  so 
these  chords  should  be  regarded  as  standing  midway  between  the  families  of  their  constituents. 
This  combination  of  two  different  elements  accounts  in  great  measure  for  the  characteristic  effect 
of  such  seventh  chords. 
/).  Section  d  of  Chapter  Twenty  applies  to  these  chords  as  well,  with  the  following  exception: 


Chapter  Twenty-four 


39 


The  sevenths  of  these  seventh  chords  may  also  resolve  one  degree  upward,  as  the  tendency 
for  such  a  resolution  seems  to  be  a  characteristic  of  the  interval  of  a  major  seventh.  The  lower 
constituent  of  such  a  seventh  chord  may  be  sustained  or  not. 


Examples : 


c.  Sections  c,  f,  g,  h  and  i  of  Chapter  Twenty  may  be  applied  here  as  well. 

(/.  The  best  forms  of  these  chord.s  are  those  of  the  root  position  and  first  inversion,  as  this  places  the 
roots  of  the  constituent  triads  in  the  Bass.  The  chord  sevenths  should  generally  be  placed  in 
the  Soprano,  and,  unless  the  sevenths  are  prepared,  the  roots  and  sevenths  should  be  at  least  a 
seventh  apart.     As  in  secondary  sevenths,  no  member  should  be  omitted. 

e.  On  account  of  the  greater  dissonance  of  these  chords,  their  sevenths  should  be  more  frequently  pre- 
pared than  those  of  the  secondary  seventh  chords. 

Example : 


nTi y^ 

£    0 

— e — 

— a 

im—  ■■ 

V      0 

V 

■9' 

h\-  ■    0 

0 

n 

0).       '^ 

/ 

0 

U g — 1 

1 1 

a 

^7       iT 


/.  Note  that  the  primary  sevenths  are  the  only  ones  which  cuiUain  tiie  leatling  tone  and  have  principal 
tones  for  both  root  and  seventh.  Note  that  secondary  seventh  chords  generally  have  principal 
tones  for  third  and  seventh  ;  note  that  this  is  not  the  case  with  tertiary  sevenths  which  have  them 
as  root  and  fifth.  .'\s  the  dissonance  of  a  seventh  throws  the  tones  composing  it  into  high  relief, 
the  facts  just  mentioned  may  account  for  the  primary  sevenths  having  the  greatest  key-deter- 
mining power  and  the  tertiary  sevenths  the  least. 


r.XliRClSE. 


.^pply  the  exercise  of  Chajiter  Twenty  to  the  tertiary  sevenths  in  various  major  keys. 

_(/.  Tertiary  sevciuhs  are  not  used  as  often  as  secondary  sevenths,  as  their  acute  dissonances  are  seldom 
needed. 


40 


Cha/iter  Twenty-five 


INVENTION. 

Compose  two  periods  in  major  keys  with  the  eflfect  of  the  various  seventh  chords  in  mind. 
Harmonize  them  according  to  the  method  given  in  Chapter  Twenty-one. 

The  materials  of  Chapters  Fifteen  and  Eighteen  may  be  applied  here  also,  as  well  as  through- 
out this  course,  but  the  following  fact  should  be  noted :  suspension  is  the  only  device  which 
always  sounds  well  in  combination  with  tertiary  sevenths,  in  fact,  there  is  little  or  no  difference 
between  a  chord  seventh  that  is  prepared  and  resolves  one  degree  downward  and  a  suspension. 
By  such  means,  all  of  our  seventh  chords  evolved  into  our  musical  vocabulary.  The  seventh  of  a 
tertiary  seventh,  which  is  prepared  and  then  resolved  upward,  has  the  effect  of  an  upward  resolv- 
ing suspension. 

The  passive  resolution  of  the  Subdominanl  seventh  makes  an  acceptable  perfect  cadence  that 
seems  somewhat  plagal  in  effect. 


Chapter  Tirenty-five. 

THE  EVOLUTION,  TREATMENT.  .^ND  .APPLICATION  OF  THE  TERTIARY  SEVENTHS   IN  THE   MINOR   MODE. 

Referring  to  Figure  49  in  Chapter  Twenty-two,  we  find  tertiary  sevenths  on  the  third  and  sixth 
degrees  of  the  minor  scale.  Note  that  the  tertiary  seventh  on  the  third  degree  of  the  minor 
scale  has  two  forms,  one  with  the  perfect  fifth  and  the  other  with  the  augmented  fifth.  The  use 
of  the  leading  tone  will  also  give  us  a  tertiary  seventh  on  the  first  degree. 

We  may  apply  the  same  theory  of  origin  as  that  given  for  the  tertiary  sevenths  in  major  keys; 
its  application  is  illustrated  below. 


^•*"  =  ^%   \%    'H'fl'rtt   C??! 


U 


Ml 


M 


1.0 


^-j\\\  ^a    |l^ 


X7  TSL-i  jr^ 

The  third  paragraph  of  Section  a  of  the  preceding  chapter  applies  here  also. 
Section   h  of   the   preceding  chapter   also   applies  here. 
Examples : 


Si/C^.  SS 


m 


s 


^ 


|h^ 


It 


^ 


^ 


^ 


^ 


Htfc,       -ET 


\ote  that  the  seventh  of  the  I,  cannot  be  led  downward,  as  it  would  give  rise  to  a  step  of 
an  augmented  second,  or  to  the  presentation  of  the  seventh  degree  of  the  scale  first  as  leading 
lone  and  then  as  natural  seventh. 


Chapter  Twenty-six 

b.  Sections  c,  d  and  e  of  the  preceding  chapter  apply  here  also. 
Examples : 


41 


c.  Observe  the  application  of  Sections  /  and  g  of  the  preceding  chapter  to  these  chords. 

EXERCISE. 

Apply  the  exercise  of  Chapter  Twenty  to  the  tertiary  sevenths  in  various  minor  keys. 
Ohtain  all  the  variety  of  effect  possible  by  using  both  the  prepared  and  the  unprepared  sev- 
enth, and  also  both  forms  of  the  IIIj. 

E.\K    TRAI.MNG. 

Be  able  to  distinguish  l^etween  the  various  forms  and  kinds  of  seventh  chords  studied  by 
name,  inversion,  and  if  possible,  by  location  of  the  seventh  of  each  chord.  Note  different 
effects  produced  by  the  two  forms  of  the  III;. 

J.WENTION. 

Work   out   the   invention   of   the   preceding  chapter  in  three  different  minor  keys. 


Chapter  Tiventy-stv. 

THE   FIRST   SPECIES  OF   MODUL.^TION — ITS  EVOLl'TION   .\ND  TRE.\TMENT. 

a.  Modulation  may  be  briefly  defined  as  the  process  of  changing  from  one  key  to  another.  The  keys 
in  question  will  be  found  to  have  the  same  relation  between  themselves  that  chords  in  the  same 
key  have  to  each  other.    This  refers  particularly  to  the  Tonic  chords  of  such  keys. 

As  two  chords  may  have  one  or  more  common  tones,  so  we  will  find  that  two  keys  have  one 
or  more  common  chords.  As  each  of  these  common  chords  occurs  in  the  different  keys,  even  if 
under  a  different  name  in  each  key,  it  may  be  used  as  a  "bridge"  for  passing  from  one  key  to 
another. 

h.  By  referring  to  Chapters  Twelve  and  I'ourtecn.  \vc  will  liml  discussions  and  illustrations  of  the 
relation  to  the  Tonic  of  five  of  the  other  triads  of  each  key.  If  these  triads  are  taken  as  the 
Tonic  triafls  of  new  keys,  the  relation  between  the.se  new  keys  and  the  original  Tonic  center  will 
be  found  to  be  the  same.  The  \TI  in  major  keys,  and  the  11  and  \'1I  in  minor  keys,  may  be 
disregarded  in  this  work,  as,  being  diminished  triads,  they  can  be  the  Tonic  triads  of  neither  a 
major  nor  a  minor  key. 

f.  By  taking  the  various  major  and  minor  triads  of  the  major  and  minor  keys  in  turn,  we  find  that  we 
may  modulate,  in  the  major  key,  to  the  keys  who.se  Tonic  triads  are  the  same  as  the  triads  on 
the  second,  third,  fourth,  fifth,  and  sixth  degrees  of  the  scale.  On  account  of  the  form  of  these 
triads,  the  keys  re])resented  by  the  triads  on  the   second,  third,  and  sixth   degrees  of  the  major 


42  Cha/iter  Tu-eiity-six 

scale  must  be  minor  keys,  and  those  keys  represented  by  the  triads  on  the  fourth  and  fifth  de- 
grees of  the  major  scale  must  be  major  keys.  For  example,  we  may  modulate  from  C  major 
to  any  one  of  the  following  keys:    D  minor,  E  minor,  F  major,  G  major,  A  minor. 

The  most  satisfactory  form  of  modulation  from  minor  keys  is  that  which  is  based  on  the 
minor  scale  with  natural  seventh.  Applying  the  principle  given  in  the  preceding  paragraph, 
we  find  that  we  may  modulate,  in  the  minor  key.  to  the  keys  whose  Tonic  triads  are  the  same 
as  the  triads  on  the  third,  fourth,  fifth,  sixth,  and  natural  seventh  degrees  of  the  scale.  On 
account  of  the  form  of  the  minor  scale  just  mentioned,  the  Y  must  be  a  minor  triad,  and  the  \TI 
a  major  triad.  On  account  of  the  form  of  these  triads  the  keys  represented  by  the  triads  on  the 
third,  sixth,  and  seventh  degrees  of  the  minor  scale  must  be  major  keys,  and  those  keys  repre- 
sented by  the  triads  on  the  fourth  and  fifth  degrees  of  the  minor  scale  must  be  minor  keys.  For 
example,  we  may  modulate  from  A  minor  to  any  one  of  the  following  keys:  C  major,  D  minor, 
E  minor,  F  major,  G  major. 

(/.  These  possible  modulations  are  grouped  under  the  title  of  "The  First  Species  of  Modulation." 

e.  We  may  illustrate  the  use  of  our  materials  and  resources  in  the  following  manner: 


%      i      g 


li    g    ti    §    ?: 


hn — w-n-^^^^^^ 


In  the  above  figure,  we  have  the  scales  of  C  major  and  D  minor,  each  scale  having  triads 
erected  upon  it  according  to  its  own  key.     The  "comnion  chords"  are  connected  by  lines. 


Figure  61  above  is  an  example  which  begins  in  the  key  of  C  major,  proceeds  to  a  chord  that 
is  common  to  both  C  major  and  D  minor,  and  then  makes  a  cadence  in  D  minor,  after  which  it 
proceeds  to  another  common  chord,  and  by  its  means,  back  to  a  cadence  in  C  major. 

/.  Note  that  the  dominant  seventh  chord  is  used  to  cstal)lish  the  key  in  which  we  begin,  to  confirm  the 
key  to  which  we  modulate,  and  to  re-establish  the  original  key  upon  our  return  to  it.  Upon  in- 
vestigation, we  will  find  that  the  dominant  seventh  of  any  key  does  not  occur  as  a  diatonic  chord 
in  any  of  the  other  keys  to  which  we  may  modulate  at  present.  Tliis  accounts  for  our  use  of  it 
as  mentioned  just  above,  in  confirming  a  key. 


Chapter  Ticenty-six 


43 


g.  Observe  from  Figure  61  tliat  tlie  same  means  employed  in  modulating  are  used  in  returning  to  the 
original  key. 

h.  Avoid  the  use  of  bridge  chords  which  would  necessitate  the  use  of  the  leading  tone  of  a  minor 
key  directly  after  its  natural  seventh,  or  vice  versa. 

INVENTION. 

By  means  of  a  table  of  common  chords  similar  to  Figure  60,  construct  examples  of  the  first 
species  of  modulation  similar  to  Figure  61.  Obtain  as  much  variety  as  possible  in  the  choice 
of  bridge  chords  and  modulations. 

These  inventions  need  not  be  in  period  form  at  present. 

(.  By  adding  a  seventh  to  each  of  the  triads  shown  in  Figure  60  various  "common  seventh  chords" 
will  be  obtained. 

INVliNTION. 

Work  out  inventions  similar  to  the  preceding  one,  making  use  of  "common  seventh  chords." 
Pay  particular  attention  to  Section  h  above. 

j.  Some  of  the  above  inventions  should  make  use  of  the  second  inversion  of  the  Tonic  triad  of  the  key 
to  which  the  modulation  is  made,  placing  it  on  an  accented  beat,  and  following  it  by  the  dominant 
seventh  of  this  new  key.  This  use  of  the  second  inversion  has  been  hitherto  forbidden  for 
secondary  triads,  as  it  promises  a  modulation  which  could  not  be  applied  until  now;  the  use 
of  such  a  second  inversion  may  be  now  ju-.tilif(l  by  ihe  work  just  taken  up. 

/.•.  "Bridge  chords"  are  not  invariably  used  in  modulation.  Providing  that  the  key  succession  is  correct, 
we  may  anticipate  a  new  key  by  placing  one  of  its  subdominant  secondary  seventh  chords 
before  the  dominant  seventh  of  the  new  key,  or  before  its  I«  followed  bv  its  dominant  seventh. 
The   same   device   may   be   used    in   returning  to  the  original  ke\'. 

Example : 


^      CX        t 


i^ 


i^^i 


7^^ 


s^ 


3£ 


r^ 


T=^ 


t^ 


T^ 


s — 


^^ 


r^^ 


^ 


fei 


S 


-C^. 


W^ 


^ 


<  r         X6.      7  X 


^  -o^^s 


^i         JC%i^     ^^ 


Section  h  al)ove  applies  here  also. 

The   use  of  tertiarv   sev(nth'~   in   the   work  of  ."section  k  is  not  to  be  recommended. 


INVENTION. 

.\pply   the   device   mentioned    iti    Section  h   in   various   inventions   similar   in    form   to   those 
just  written. 


44 


Chapter  Twenty-seven 


Chapter  Tiventy-seven. 

THE    DOIBI.K    PERIOD.       "FREE    RESOLUTION"    OF   SEVENTH    CHORDS. 


u.  A  double  period  consists  of  four  phrases,  and  may  he  regarded  as  a  development  of  the  simi)le  period 
already   familiar. 

Particular  attention  should  be  paid  to  the  treatment  of  the  cadences.  The  cadence  at  the 
end  of  the  second  phrase  should  generally  be  a  half-cadence  of  considerable  rhythmic  weight 
and  effect,  in  order  to  terminate  the  first  half  of  the  double  period,  but,  at  the  same  time,  not  to 
bring  it  to  a  full  close.  The  cadences  at  the  ends  of  the  first  and  third  phrases  are  generally 
half-cadences  of  less  rhythmic  weight  and  prominence  than  that  at  the  end  of  the  second  phrase. 
They  may,  however,  occasionally  employ  cadences  to  the  first  inversion  of  the  Tonic  chord,  or 
to  the  root  position  or  first  inversion  of  the  \'l. 

A  fine  example  of  the  form  of  the  double  period  will  be  found  given  by  the  first  sixteen 
measures  of  Beethoven's  Sonata  for  the  Piano,  Opus  26.  Other  examples  of  the  double  period 
should  be  looked  up,  taking  this  one  for  a  model. 

b.  Note  that  the  first  and  third  phrases  of  the  double  period  are  always  alike,  except  jjossil^ly  toward 

their  ends,  although  the  third  phrase  sometimes  gives  the  material  of  the  first  in  a  more  orna- 
mented form. 

The  second  and  fourth  phrases  may  be  more  or  less  alike  as  in  the  example  we  just  quoted: 
their  ends  must,  of  course,  be  somewhat  ditTcrent  on  account  of  the  different  cadences  necessary. 

Double  periods  will  frequently  be  found  in  which  the  secotid  and  fourth  phrases  are  quite 
dissimilar,  and  such  forms  should  be  construcU'(l  by  the  student,  but  all  true  double  periods  will 
have  the  beginnings,  at  least,  of  their  first  and  third  phrases  alike  or  very  similar. 

INVENTION. 

Expand  several  previously  written  inventions  into  double  periods.  Obtain  as  much  variety 
as  possible  in  the  treatments  given  the  second  and  fourth  phrases. 

c.  The  treatment  given  the  dominant  sevcutli  in  the   jircceding  i  hnplers   i^  the  only  one   possible,  on 

account  of  the  strong  progressive  tendency  of  its  third  and  seventh.  As  the  secondary  seventh 
chords  do  not  tend  to  progress  so  strongly,  certain  freer  treatments  of  their  sevenths  have  been 
in  use  since  the  time  of  Haydn.  These  freer  treatments  are  known  as  "Free  Resolutions." 
Thev  are  possible  with  both  secondary  and  tertiary  sevenths,  but  are  generally  to  be  preferred 
with  the  former  when  opportunity  for  them  is  encountered.  We  have  two  types  of  free  active 
resolution  and  two  types  of  free  passive  resoliitinn.  which  arc  here  illustrated  and  discussed. 


d.  In  all  of  these  treatments  note  tliat  the  chord  succession  is  correct, 
even  with  free  treatments. 


his  must  be  always  the  case, 


•So..  63 


^ 


m 


^7 


Chapter  Twenty-seven 


45 


In  the  above  figure,  note  that  the  seventh  of  the  first  chord  is  taken  by  the  Alto  and  its  tone  of 
resoUition  is  taken  just  below  it  by  the  Soprano  of  the  following  chord.  This  is  the  first  type 
of  free  active  resolution,  in  which,  as  the  chord  seventh  is  obliged  to  move,  it  falls  to  a  note  of 
the  following  chord  while  another  voice  of  the  following  chord  takes  the  tone  of  resolution  at 
the  same  level  that  the  seventh  itself  would  have  taken,  had  the  seventh  resolved  in  the  usual 
strict  fashion. 

This  type  of  resolution  will  be  found,  upon  experiment,  to  be  efTcctive  when  the  seventh  is 
taken  by  the  Alto  or  Tenor. 


^^.iH- 

-p 



[ai     a 

—Q^ 

—-Q, 

■&■ 

o 

—5 — 

-ev^ — s — 

a 

-^ Q— 

—0 — 

— Q— 

%1    %i    I 

/'.  In  the  above  figure,  note  that  the  seventh  of  the  first  chord  is  taken  by  the  Tenor  and  its  tone  of  reso- 
lution is  taken  a  ninth  below  it  by  tlie  Bass  of  the  following  chord.  This  is  the  second  type  of 
free  active  resolution  in  which,  as  the  chord  seventh  is  obliged  to  move,  it  falls  to  a  note  of  the 
following  chord,  while  another  voice  of  the  following  chord  takes  the  tone  of  resolution  at  the 
distance  of  over  an  octave  below  it. 

This  type  of  resolution,  like  the  previous  type,  is  most  effective  when  the  seventh  is  taken 
by  the  Alto  or  Tenor. 


(/.  In  the  above  figure  note  that  the  seventh  of  the  lirst  chord  is  taken  by  the  .\lto,  which  leaps  down, 
allowing  the  Soprano  to  passively  resolve  the  .seventh  by  taking  it  on  the  same  /czv/  in  the  fol- 
lowing chord.    This  is  the  first  type  of  free  passive  resolution. 

As,  in  passive  resolution,  the  chord  seventh  is  repented,  the  voice  which  takes  the  seventh 
may  move  either  up  or  down. 

This  type  of  resolution  may  occur  when  the  seventh  is  taken  by  the  Soprano,  Alto,  or  Tenor. 


46 


Chapter  Twenty-ei^ht 


h.  In  the  above  figure,  note  that  the  seventh  of  the  first  chord  is  taken  by  the  Tenor,  and  that 
this  note  is  taken  by  the  Soprano  in  the  next  chord  on  a  different  level.  This  is  the  second  type 
of  free  passive  resolution. 

As  this  also  is  passive  resolution,  in  which  the  seventh  is  repeated,  the  voice  which  takes  the 
seventh  may  move  either  up  or  down  and  the  repetition  of  the  seventh  in  the  next  chord  may  be 
at  either  a  higher  or  a  lower  level. 

This  type  of   resolution  may  occur   when  the  seventh  is  taken  by  the  Soprano,  Alto,  or 
Tenor  as  before, 
t.  When  the  seventh  of  a  chord  appears  in  the  Bass,  free  resolution  will  be  found  difficult. 

LXERCISE. 

Write   various    examples    of   the    different  kinds  of  free  resolution  in  various  chords  and  keys. 
These  may  be  combined,  with  great  profit,  in  the  work  on  the  double  period,  as  they  permit 
much  more  freedom  in  the  handling  of  melodies. 


Chapter  Twenty-ei^ht. 

THE  EVOLUTION,  TREATMENT,  AND  APPLICATION  OF  THE     PRIMARY     NINTH     CHORD,     AND     THE     SEVENTH 
CHORD  ON  THE  LEADING  TONE,  IN  MAJOR   AND    MINOR    KEYS.      FIVE   PART    HARMONY. 

a.  With  the  exception  of  the  seventh  chord  on  the  leading  tone  in  the  major  and  the  minor  key,  which 
will  be  explained  later,  the  derivation  and  use  of  all  the  diatonic  seventh  chords  has  been  treated 
of  in  the  preceding  chapters.  Further  harmonic  material  must  consist  of  chords  of  five  different 
notes. 


We  will  investigate  the  following  series  of  overtones  in  the  effort  to  find  such  a  chord. 


f:^.(57 


P 


^ 


tto 


p(^V^        ^f  t^f    f  .,. 


^ 


* 


h.  The  above  figure  contains  the  most  extended  series  of  overtones  needed  for  our  studies  in  diatonic 
harmony.    The  following  remarks  may  be  made  about  it  before  going  further: 

Note  the  double  "accidental"  before  the  tenth  and  twelfth  overtones.  These  notes  are  so 
modified  because  their  sounds  stand  between  the  two  tones  implied  by  the  two  "accidental" 
signs.  The  eleventh,  thirteenth  and  fifteenth  overtones  are  duplications  of  tones  already  found 
in  the  series ;  the  fourteenth  overtone  might  be  used  to  explain  the  origin  of  such  chords  as  the 
Ij,  but,  upon  further  consideration,  such  a  treatment  of  this  overtone  will  be  seen  to  be  arbitrary 
and   far-fetched. 

c.  We  may  obtain  a  chord  of  five  different  notes  from  the  fundamental  and  the  first  eight  overtones 
(first  to  ninth  partials),  which  may  be  reduced  to  the  following  form. 

H  t,  a    II 


Chapter  Twenty-eight  47 

On  account  of  the  interval  of  a  ninth  between  the  root  and  top  note  of  this  chord,  it  is 
called  a  "Chord  of  the  Ninth."  ^\■e  may  describe  this  chord  by  saying  that  it  consists  of  a  major 
triad,  minor  seventh,  and  major  ninth. 

d.  Like  the  dominant  seventh,  this  chord  will  be  seen  to  be  possible  as  a  diatonic  chord  on  the  fifth 
degree  only  of  the  major  scale.  We  may  call  this,  therefore,  "The  Dominant  Chord  of  the 
Ninth."  On  account  of  its  natural  origin,  it  is  the  only  primary  ninth  chord  in  the  diatonic  key- 
scheme. 

c.  In  our  previous  comparisons  of  the  series  of  overtones  and  undertones,  we  have  found  that  the 
undertones  were  built  downward  by  the  same  series  of  intervals  with  which  the  overtones  were 
built  upward.  This  has  been  found  to  hold  good  in  the  more  extended  series,  so  a  table  of 
undertones  as  complete  as  the  overtone  table  of  Figure  67  may  be  built.  By  taking  the  gen- 
erator and  the  first  eight  undertones,  we  find  that  we  can  get  an  undertone  ninth  chord  of 
exactly  the  same  form  as  I'igure  68.  and,  by  investigation,  we  will  find  that  this  is  a  possible - 
undertone  chord  from  the  first  degree  of  the  modern  minor  scale.  The  chord  is  not  used  in 
the  minor  key,  as  we  not  only  think  it  ui)ward,  but  associate  it  entirely  with  the  relative  major 
key.  The  latter  reason  forbids  our  use  of  this  chord  as  an  overtone  ninth  chord  on  the  natural 
seventh  degree  of  the  minor  scale,  except  upon  rare  occasions. 

In  actual  practice,  we  use  this  chord  as  an  overtone  chord  on  the  Dominant  of  the  minor 
scale,  lowering  its   ninth   to   make   it   conform  to  the  kev. 

/.  Note  that  the  dominant  ninth  chord  determines  not  only  the  key,  but  the  mode,  as  may  be  seen  from 
a  com])arison  of  the  dominant  ninths  of  C  m;ijor  and  t.'  minor. 

(J.  We  may  regard  and  treat  the  dominant  ninths  of  the  major  and  minor  keys  alike.  They  may  be  re- 
garded as  a  combination  of  the  dominant  seventh  with  a  major  or  a  minor  ninth,  according  to 
key.  The  root,  third,  and  seventh  of  this  dominant  seventh  are  to  be  treated  as  before;  as  the 
ninth  is  a  dissonant  active  tone  it  should  be  invariably  led  one  degree  downward  .  Because  of 
this  resolution  of  the  niiuh,  the  fifth  of  the  chord,  if  present  below  the  ninth,  should  be  led  to 
the  third  of  the  scale  in  the  following  chord,  to  avoid  consecutive  fifths  with  the  ninth.  If  the 
fifth  is  above  the  ninth,  it  may  go  to  the  first  degree  of  the  .scale  as  before. 

//.  .\s  the  dominant  clinid  is  a  pi  iin.u  \  <li><si)naiicc.  it  is  too  clnse  to  liie  Tonic  family  to  passively 
resolve  its  ninth. 

/.  The  ninth  may  lie  in  any  ])art  but  the  I'.ass.  in  order  to  niaint.iin  its  true  character,  and.  for  (he  same 
reason,  must  be  always  at  least  a  ninth  above  the  root. 

Subject  to  these  restrictions,  the  dominant  seveiUh  eieinein  of  this  chord  may  be  freely  inverted. 

j.  In  four  i)art  harmony,  the  fifth  is  generally  omitted,  as  we  can  do  best  without  it.  Occasionally  one 
finds  domin;mt  ninth  chords  with  third  omitteil   and   fifth  present. 

/;.  In  ail  forms  of  the  chords  of  the  ninth,  and  the  chords  of  the  eleventh  ami  thirteenth  which  are  to  be 
studiefl  later,  the  sevenths  of  the  chords  are  necessary  for  good  effect. 


48 


Chapter  Tnetity-eight 


Examples : 


^.fo<? 


S 


V 


^ 


■5 


z 


■e- 


tt^. 


I.  This  chord,  like  the  dominant  seventh,  requires  no  preparation. 

The  ninth  may  be  resolved  before  the  seventh,  but  not  after  it,  except  where  the  effect  of 
suspension  is  required. 

m.  The  ninth  may  fall  to  some  tone  of  the  dominant  seventh  chord,  achieving,  in  a  sense,  a  sort  of 
free  active  resolution  of  the  second  type.  The  chord  may  be  preceded  by  the  \\  or  by  any 
chord  that  can  precede  the  V;. 


Examples : 


^.70 


n.  Like  the  dominant  seventli  chord  the  dominant  ninih  ma\-  appear  wiihout  its  root.  This  mav 
account  for  the  origin  of  the  seventh  chord  on  the  leading  tone.  Such  a  chord,  in  the  minor  key, 
is  called  "The  Chord  of  the  Diminished  Seventh." 

(I.  The  treatment  of  each  tone  of •  this  chord  is  the  same  as  that  employed  when  the  fundamental  is 
I^resent. 

On  account  of  the  absence  of  the  fundaniciital,  the  ninth  mav  occur  in  the  Bass. 

Examples : 


J^ 

-fe n — 

0 

0 1 

0 

rt 1 

— '^ — 

0 

—0 — 
-Or 

0 

5 
2 

—0 — 
-0- 

0 

-0- 

— & — 

0 

— & — 

-^. 

— rt 

— Q 

— S— 

—% — 

— 0 — 

— t^ 

=^=^ 

0 

0 

0 

— & — ^ 

0 

ft 


Chapter  Twenty-ei£hi  49 

KAK    TRAINING. 

Learn  to  recognize  the  difference  between  the  dominant  ninth  chords  with  and  without  root 
in  major  and  minor  modes.  If  possible,  be  able  to  distinguish  these  chords  by  inversion  as  well 
as  by  the  position  of  the  seventh  and  ninth. 

Note  the  similarity  in  form  between  the  dominant  ninth  without  root  in  a  major  key  and  the 
supertonic  seventh  in  its  relative  minor  key.  Compare  this  fact  with  the  actual  difference  in 
the  effects  and  treatments  of  these  chords. 

LXERCISE. 

Write  the  above  four  chords  in  various  inversions  and  keys,  with  various  combinations  of 
prepared  and  unprepared  dissonance.     Resolve  each  chord. 

Apply  these  chords  in  some  exercises  in  modulation,  as  in  Chapter  Twenty-six,  for  confirming 
the  keys.     Note  that  those  with  root  are  the  most  effective  for  this. 

INVENTION. 

Write  two  double  periods  in  major  keys,  and  two  in  minor  keys,  using  these  chords. 
p.  As  members  of  cadences,  the  dominant  ninth  chords  with  root  are  to  be  preferred  to  those  without 
root. 

(].  FIVE  P.\RT  H.XRMONV.     P'ive  part  harmony  is  particularly  valuable  for  the  treatment  of  complicated 
chords  or  the  attainment  of  sonorous  effects. 

The  only  new  directions  which  are  needed  are  those  concerning  doubling.  The  old  rules  for 
doubling  are  generally  valid. 

r.  Any  member  that  may  be  doubled  in  four  part  harmony  may  be  tripled  in  five  part  harmony.     The 
application  of  this  fact  allows  wider  choice  of  effects  in  the  treatment  of  dissonant  chords. 

When  necessary,  the  thirds  of  principle  chords  may  be  doubled  if  they  can  enter  smoothly, 
but  this  should  be  avoided  whenever  possible,  particularly  with  regard  to  the  leading  tone. 

Chord  fifths  may  be  doubled.  .\  chord  may  consequently  appear  with  both  its  root  and  fifth 
doubled. 

Six-four  chords  may  triple  their  fifths  or  double  l)oth  root  and  fifth,  or  the  second  inversion 
of  secondary  triads  may  doulile  tiicir  fifths  and  thirds. 

We  may  double  the  leading  tone  and  the  root,  or  the  third,  in  the  second  inversion  of  the 
III  in  major  keys,  but  the  leading  tone  should  never  be  tripled. 

The  triad  on  the  leading  tone  may  double  both  its  third  and  fifth,  or  triple  its  third.  .\n 
augmented   triad   siiould   f,'cncrally   (loublc   its  root  and  lliini,  oi-  Iriplf  its  tliird.  in  ail  jiositions. 

With  the  exception  of  the  triad  on  the  Icinling  tone,  no  chord  seveiUh.  ninth,  eleventh,  or 
tiiirteenth  should  be  ever  doubled. 

Secondary  and  tertiary  seventh  chords  m.i\  doulilc  iluir  thirds  wlu-n  convenient,  as  these  are 
the  roots  of  constituent  triads. 
s.  The  choice  between  different  doublings  or  between  doubling  and  tri|)liiif;  will  flc|iend  on  the  require- 
ments of  good  voice  leading. 
/.   Xinth  chords  may  occur  complete,  or  with  any  of  the  doublings  or  triplings  mentioned  above. 

When  the  fifths  of  seventh  and  ninth  cIkikIs  occur  in  the  Piass,  the  best  sonority  is  obtained 
by  doubling  them. 

INVENTION. 

Select  various  inventions  done   from  Chapters  I'ifteen  onward  and  rewrite  them  in  five  part 

harmouv. 


50 


Chapter  Tiventy-nine 


Chapter  Twenty-nine. 


THE    EVOLUTION,    TREATMENT,    AND    APPLICATION    OK    THE    .SECONDARY,    AND    TERTIARY    CHORDS    OK    THE 

NINTH    IN    MAJOR    AND     MINOR     MODES. 

a.     By  the  addition  of  a  third  to  each  of  the  seveiitli  chords  in  a  major  and  a  minor  key,  we  will  ohtain 
the  following  series  of  ninth  chords : 


fc«^.l% 


^^^ 


S 


^  \%  4 


%W% 


H  H 


...  ,^t  ^  ^1  ,^!  i^t  ^.fe 


I 


E 


As  the  dominant  ninth  chords  have  alrea(i\'  heen  discussed,  we  will  turn  to  the  studv  of  the 
Others. 

b.  W'e  will  find  that  the  ninth  chord  on  the  natural  seventh  degree  of  the  minor  scale  is  the  only  other 

one  made  from  the  overtones  of  its  root  or  from  the  undertones  of  the  ninth.     This  chord  was 
discussed  in  Section  c  of  the  previous  chapter. 

c.  With  regard  to  the  other  ninth  chords,  we  may  classify  them  according  to  the  qualitv  of  their  ninths, 

and  call  those  chords  which  have  major  ninths  "secondary  ninth  chords,"  and  those  which  have 

minor  ninths  "tertiary  ninth  chords." 
(/.  The  extreme  dissonance  of  the  tertiary  ninths  often  renders  them  unmusical  unless  their  ninths  are 

prepared  and  resolved  like  suspensions,  or  unless  the  chords  seem  to  arise  as  accidental  formations 

of  a  "passing"  character.     They  are  not  recommended  for  use  at  present. 
e.  The  secondary  ninth  chords  may  be  regarded  as  a  union  of  two  seventh  chords,  as  illustrated  below. 


0 


rz- 


Those  secondary  ninth  chords  whose  lower  constituent  seventh  chords  are  .secondary  sevenths 
will  be  found  to  be  more  useful  and  of  more  agreeable  sound  than  other  formations  of  such 
chords.  These  chords  may  be  preceded  and  followed  by  chords  which  would  make  good  pro- 
gression with  either  of  their  constituent  seventh  chords.  This  will  also  apply  to  the  introduction 
of  any  one  of  these  chords  by  either  of  their  constituent  seventh  chords,  or  by  any  of  the  con- 
stituent triads  of  these  seventh  chords. 

/■.  These  chords  require  preparation  of  their  sevenths  and  ninths  more  frequently  than  the  similar  inter- 
vals of  primary  ninths,  but  they  do  not  alwa}-s  demand  it. 


Chapter  Tiventy-nine 


51 


In  resolving  these  chords,  the  root  and  seventh  of  each  constituent  seventh  chord  may  lie 
treated  in  any  of  the  various  ways  previously  used ;  that  is,  for  instance,  the  seventh  of  one  of 
the  constituent  chords  may  take  one  kind  of  resolution  while  the  seventh  of  the  other  constituent 
seventh  chord  takes  some  different  kind. 

g.  These  chords  best  occur,  in  five  part  harmony,  with  the  root  of  one  of  their  constituent  sevenths  in 
the  Bass.    In  four  part  harmony,  the  chord  fifth  is  the  best  note  to  omit. 


Examples : 


^.7V 


5 


S 


^ 


^ 


:saz 


ST  %<{  'Sr<\ 


«1 


""i 


It    .Avoid  leading  the  interval  of  a  ninth  to  the  interval  of  an  octave  by  a  movement  of  the  lower  note 
of  the  ninth. 

The  ninth  may  fall  to  some  interval  of  the  lower  constituent  seventh  chord,  but  no  secondary 
ninth  chord  should  be  led  to  its  upper  constituent  seventh. 

I.  The  ninth  should  be  kept  in  one  of  the  upper  parts,  the  Soprano  being  preferred,  and  must  be  always 
at  least  a  ninth  above  the  root  of  the  lower  constituent  seventh  chord. 

j.  The  student  is  recommended  to  the  study  of  the  works  of  Grieg  in  which  may  be  found  fine  exam- 
]iles  of  the  treatment  of  these  chords. 

r..\R    TU.MNING. 

Learn  to  recognize  the  difference  of  effect  between  various  varieties  of  these  chords  and  the 
dominant  chords  of  the  ninth.  This  may  be  extended  to  include  recognition  of  the  position  of 
the  roots  and  sevenths  of  tlie  constituent  seventh  chords. 

EXERCISE. 

Write  various  examples  of  different  forms  of  introduction  and  re.solution  of  secondary  nintii 
chords  in  major  and  minor  keys.  Get  as  much  variety  as  possible  in  the  treatment  of  the  con- 
stituent seventh  chords. 


IXVE.MTION. 


Construct   two  doulile   periods,   using  a    few  secondary  ninth  chords  in  each  one. 
These  chords  should  not  he  used  often.  btU  are  sometimes  effective,  particularly  when  given 
free  resolution. 


52 


Chapter  Thirty 


Chapter  TJiirty. 

THE   EVOLUTION'   AND  APPLICATION'  OF  THE  CHORDS  OF  THE   ELEVENTH   AND  THIRTEENTH    IN   BOTH    MAJOR 

AND    MINOR    MODES. 

<;.  We  may  obtain  a  chord  with  six  different  notes  from  the  fundamental  and  the  first  ten  overtones. 
(See  Figure  67.)  These  tones,  after  duplications  have  been  omitted,  will  take  the  following 
form : 


The  distance  between  the  root  and  top  note  will  justify  us  in  calling  this  a  "Chord  of  the 
Eleventh."  As  the  eleventh  is  out  of  key,  we  may  alter  it  to  fit  the  major  or  the  minor  key,  with 
the  following  result,  in  which  it  will  be  seen  that  the  minor  ninth  is  used  in  the  minor  key. 


£ 


■9- 

/'.  Like  the  dominant  seventh  and  dominant  ninth,  this  chord  will  be  seen  to  be  possible  as  a  diatonic 
chord  on  the  fifth  degree  only  of  the  major  or  minor  scale.  We  may  call  this,  therefore,  "The 
Dominant   Chord  of  the   Eleventh." 

c.  As  this  is  an  extremely  artificial  form,  we  do  not  apply  theories  of  undertones  to  this  chord.    Often 

as  not  the  occurrence  of  the  chord  is  more  easily  analyzed  upon  a  contrapuntal  basis. 

d.  The  extreme  dissonance  of  the  third  against  the  eleventh  causes  us  to  omit  the  third  for  the  present 

in  all  treatments  of  the  chord,  so  that  the  chord  can  appear  in  five  part  harmony  with  root, 
fifth,  seventh,  ninth,  and  eleventh,  with  doubled  root,  seventh,  ninth  and  eleventh  being  present, 
or  with  doubled  root,  fifth,  seventh,  and  eleventh.  In  four  part  harmony,  the  chord  generally 
appears  with  root,  fifth,  seventh,  and  eleventh,  or  with  root,  seventh,  ninth,  and  eleventh. 

c.  The  eleventh  should  generally  occur  in  the  Soprano,  and  may  be  prepared  or  not. 

The  other  intervals  of  the  chord  may  be  treated  as  they  were  in  the  dominant  chord  of  the 
ninth.  Consequently  the  eleventh  may  resolve  one  degree  downward,  may  fall  to  a  note  of  the 
dominant  seventh  or  ninth  chord,  or,  as  it  is  a  member  of  the  resolving  chord,  may  resolve 
passively. 

Examples : 


^■77 


3r// 


Chapter  Thirty  53 

/.  This  chord  may  be  introduced  in  the  same  manuei  as  the  dominant  ninth  chord.  The  dominant 
ninth  chord  also  may  precede  it. 

g.  There  is  often  no  difference  between  the  eleventh,  which  is  prepared  and  then  resolved  one  degree 
downward,  and  a  suspension. 

h.  The  dominant  eleventh  chord  has  been  found  to  require  the  omission  of  its  third.  If  its  root  is 
omitted  as  well,  the  effect  of  the  eleventh  is  completely  lost,  as  the  chord  becomes  a  super- 
tonic  seventh. 

i.  Secondary  and  tertiary  elevenths  may  be  said  to  exist.  They  may  be  analyzed  and  treated  like  sec- 
ondary and   tertiary   ninth  chords,   but  their  use  is  not  recommended  at  present. 

EXERCISE. 

Construct  various  examples  of  the  introduction  and   treatment  of  dominant   eleventh  chords 
in  various  major  and  minor  keys,  obtaining  variety  by  the  means  suggested  in  Section  d. 

j.  THE  CHORD  OF  THE  DOMiN.'VNT  THIRTEENTH.  From  Figure  67  of  Chapter  Twenty-eight,  we  may 
obtain  a  chord  containing  every  degree  of  the  scale,  which  may  be  condensed  for  present  pur- 
poses to  the  following  form : 


^■lU),^^ 


iMi 


^ 


The  distance  between  the  root  and  top  note  will  justify  us  in  calling  this  the  "Chord  of  the 
Thirteenth."  As  the  Thirteenth  is  out  of  key,  we  may  alter  it  to  fit  the  major  or  minor  key 
with  the  following  result: 

%.77 


'^    '7    ^  f.^ 


1^ 


k.  Like  the  dominant  chords  of  the  seventh,  ninth,  and  eleventh,  this  chord  will  be  seen  to  be  possible 
as  a  diatonic  chord  on  the  fifth  degree  only  of  the  major  and  minor  scale.     We  may  call  this 
therefore  "The  Dominant  Chord  of  the  Thirteenth." 
Section   c  of  this  chapter  applies  to  this  chord  as  well. 

/.  This  chord  generally  omits  its  fifth,  as  a  note  that  is  generally  useless  here.  The  thirteenth  is  gen- 
erally best  in  one  of  the  two  upper  parts,  preferably  the  Soprano,  although  it  occasionally 
appears  in  the  Alto,  with  the  doubled  root  above  it.  The  eleventh  may  be  used  with  the  thir- 
teenth, in  which  case  the  eleventh  is  generally  taken  by  the  Alto  and  the  thirteenth  by  the 
Soprano.     When  the  clevenlb  is  used,  the   third  must  be  omitted. 

Other  details  of  tnalnuni  may  be  applied  as  they  were  applied  to  the  eleventh,  in  Sections 
d.  e.  f  and  </.  The  thirteenth  may  also  fall  or,  rarely,  rise  to  the  first  degree  of  the  scale  or  rise 
or  fall  to  the  fifth  degree  of  the  scale  when  these  int<Tvals  occur  in  the  chord  to  which  it  resolves. 


54 

Examples : 


^.^0 


g 


5 


\s^ 


^ 


^ 


^ 


Chapter  Thirty 


III.  The  dominant  thirteenth  chord  may  occur  without  its  root.  In  this  form,  the  third  must  be  always 
present,  and  the  eleventh  consequently  omitted,  or  else  the  chord  becomes  the  supertonic  ninth. 
The  other  intervals  of  the  chord  are  treated  as  before. 


Examples : 


^^.Sl 


£     " 

— o — 

0 

0 

0 

— i? — 

o 

—0 — 

>^ 

— ^ 

0 

■0- 

■  -  0 — 

^      0 

a 

-^ 

^^ 

//.  All  thirteenth  chords  in  a  key  will  be  seen  to  have  the  same  constituent  tones  (namely,  all  seven 
degrees  of  the  scale),  but  by  rearrangement,  spacing  and  selection  of  the  tone  placed  in  the 
Bass,  the  effect  gained  may  justify  a  classification  similar  to  that  discus.sed  in  Section  i  in  regard 
to  eleventh  chords. 

EXERCISE. 

Construct  various  examples  of  the  introduction  and  treatment  of  dominant  thirteenth  chords 
in  various  major  and  minor  keys,  obtaining  variety  by  the  means  suggested  in  Section  /. 

0.  The  use  of  the  fifth  of  the  dominant  thirteenth  chord  may  be  occasionally  allowed  in  five  part  har- 
mony when  the  fifth  is  in  the  Bass. 

/>.  Note  that  the  dominant  thirteenth  may  form  a  deceptive  cadence  by  proceeding  to  the  \'l  or  the  \'I-. 
Example : 


i 


Chapter  TluTty  55 

F.AK    TRAIN  INC.. 

Re  able  to  distinguish  between  the  different  formations  of  the  dominant  eleventh  and  domi- 
nant thirteenth  chords. 

INVENTION. 

Re-harmonize  two  previously  written  double  periods,  using  various  forms  of  the  dominant 
eleventh  and  dominant  thirteenth,  both  during  the  phrases,  and,  where  possible,  at  cadences. 

New  double  periods  should  be  constructed  in  which  opportunities  are  taken  to  introduce  the 
effects  possible   with  these  chords. 

ij.  By  reviewing  the  various  forms  of  chords  studied,  we  w'ill  see  that  they  can  all  be  expressed  as 
structures  of  thirds.  If  a  third  is  added  above  or  below  a  thirteenth  chord,  it  will  be  seen  to 
merely  duplicate  a  note  already  present,  so  we  may  regard  thirteenth  chords  as  the  most  complex 
form  possible  to  diatonic  chords  which  are  built  up  by  thirds. 

Chords  built  up  by  other  intervals  such  as  fourths  and  fifths  or  by  seconds  and  sevenths  will 
be  met  with  in  modern  music.  A  few  of  these  forms  may  be  also  analyzed  as  incomplete  chords 
of  the  ninth,  eleventh,  or  thirteenth,  a  few  others  are  passing  formations  arising  from  suspension, 
but  the  majority  are  due  to  methods  of  chord  construction  which  represent  the  result  of  studies 
much  further  advanced  than  those  covered  by  the  limitations  of  diatonic  harmony. 

With  the  exception  of  the  last  noted  formations,  our  studies  in  diatonic  harmony  may  be 
regarded  as  complete. 


End  of  \'olume  I. 


^ 


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